The point(s) on the curve y = x²/6 closest to the point (0,0) are (0,0) and (±√2, 2/3).
To find the point(s) on the curve y = x²/6 closest to the point (0,0), we can use the distance formula between two points:
d = √((x₁ - x₂)² + (y₁ - y₂)²)
where (x₁, y₁) is a point on the curve and (x₂, y₂) is the point (0,0).
We want to minimize the distance d, which is equivalent to minimizing d². Therefore, we can minimize:
d² = (x₁ - 0)² + (y₁ - 0)²
= x₁² + y₁²
subject to the constraint that the point (x₁, y₁) is on the curve y = x²/6.
Substituting y = x²/6 into the expression for d², we get:
d² = x₁² + (x₁²/6)
= (7/6)x₁²
To minimize d², we minimize x₁². Since x₁² is always non-negative, the minimum occurs when x₁² = 0 or when the derivative of d² with respect to x₁ is zero.
Taking the derivative of d² with respect to x₁, we get:
d²/dx₁ = (7/3)x₁
Setting this equal to zero, we get x₁ = 0.
Therefore, the point (0,0) is one of the closest points on the curve to the point (0,0).
To find the other closest point(s), we can solve y = x²/6 for x² and substitute it into the expression for d²:
x² = 6y
d² = 7x²/6 = 7y
Therefore, to minimize d², we need to minimize y. Since y is always non-negative, the minimum occurs when y = 0 or when the derivative of d² with respect to y is zero.
Taking the derivative of d² with respect to y, we get:
d²/dy = 7
Setting this equal to zero, we get y = 0.
Substituting y = 0 into y = x²/6, we get x = 0. Therefore, the point (0,0) is one of the closest points on the curve to the point (0,0).
To find the other closest point, we can solve y = x²/6 for x:
x² = 6y
x = ±√(6y)
Substituting this into the equation for y, we get:
y = (√(6y))²/6 = 2/3
Therefore, the other closest points are (±√2, 2/3).
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Brainliest for the one who answers first!
The measure of an angle is 90.4°. What is the measure of its supplementary angle?
Answer:
89.6°
Step-by-step explanation:
180-90.4=89.6°
Answer: 89.6
Step-by-step explanation:
Supplementary means angles adding up to 180
180-90.4 = x
x=89.6 this is your supplemental angle
By about how much will g(x,y,z) = 3x + x COS Z-y sin z+y change if the point P(x,y,z) moves from P0(1.-3,0) a distance of ds= 0.1 unit toward the point P1(-1,-1,2)?
So the estimated value of √6.02 using differentials is approximately 2.4556.
The change in g(x,y,z) can be estimated using partial derivatives and differentials.
We can start by finding the partial derivatives of g(x,y,z) with respect to x, y, and z:∂g/∂x = 3 + cos(z)∂g/∂y = -sin(z) + 1∂g/∂z = -x sin(z) - y cos(z)Next, we can use the point P0(1,-3,0) and the distance ds = 0.1 to find the differentials dx, dy, and dz:dx = -2/√6 dsdy = 2/√6 dsdz = 1/√6 dsUsing these values, we can estimate the change in g:Δg ≈ (∂g/∂x) dx + (∂g/∂y) dy + (∂g/∂z) dzΔg ≈ (3 + cos(0)) (-2/√6 ds) + (-sin(0) + 1) (2/√6 ds) + (-1 sin(0) - (-3) cos(0)) (1/√6 ds)Δg ≈ (3 - 2/√6) dsPlugging in ds = 0.1, we get:Δg ≈ (3 - 2/√6) (0.1)Δg ≈ 0.389
Therefore, the change in g(x,y,z) is estimated to be approximately 0.389 units if the point P(x,y,z) moves from P0(1,-3,0) a distance of ds = 0.1 unit toward the point P1(-1,-1,2).
Suppose we want to estimate the value of √6.02 using differentials. We can start by choosing x = 6 and Δx = 0.02. Then, we need to find the derivative of f(x) = √x with respect to x:
f(x) = √x
f'(x) = 1/(2√x)
Using these values, we can estimate Δy:
Δy ≈ dy = f'(x) Δx
dy ≈ (1/(2√6)) (0.02)
dy ≈ 0.005
This means that a small change of 0.02 in x produces a small change of approximately 0.005 in y. To estimate the value of √6.02, we can add this change to the known value of √6:
√6.02 ≈ √(6 + 0.02) ≈ √6.04 ≈ 2.4556
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Suppose that someone is offering to sell you raffle tickets. there are blue, green, yellow, and red tickets available.
each ticket costs the same to purchase regardless of color. the person selling the tickets tells you that 369 blue tickets,
488 green tickets, 523 yellow tickets, and 331 red tickets have been sold. at the drawing, one ticket of each color will
be drawn, and four identical prizes will be awarded. which color ticket would you buy? explain your answer.
Buy yellow or green ticket for best chance of winning.
Which color ticket to buy?To determine which color ticket to buy, we need to calculate the probability of winning with each color.
First, we need to determine the total number of possible combinations of one ticket of each color. This can be calculated by multiplying the number of tickets of each color together:
Total number of possible combinations = 369 x 488 x 523 x 331 = 39,123,833,144
Next, we need to determine the probability of winning with each color. Since there are four identical prizes, the probability of winning with any one color is the same. We can calculate the probability of winning with a specific color by dividing the number of tickets of that color by the total number of possible combinations:
Probability of winning with blue ticket = 369 / 39,123,833,144 ≈ 0.00000943
Probability of winning with green ticket = 488 / 39,123,833,144 ≈ 0.00001248
Probability of winning with yellow ticket = 523 / 39,123,833,144 ≈ 0.00001337
Probability of winning with red ticket = 331 / 39,123,833,144 ≈ 0.00000846
From the calculations above, we can see that the highest probability of winning is with the yellow ticket, followed closely by the green ticket. Therefore, if we want to maximize our chances of winning one of the prizes, we should buy the yellow or green ticket.
It's important to note that the difference in probabilities between the yellow and green tickets is very small, so the decision of which color to choose ultimately depends on personal preference.
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1. For the solid bounded by the panes 2 = 1 - x and z=1-y in the first octant (which is the same as being bounded by x = 0, y = 0, 2 = 0), one triple integral that describes the volume of the solid is: 1- SL | 1 d:dyds + C5 .** 1 dzdyudar 0 lo z , Z=1-4 ។ Z=1- х Find three other orders of integration that describe this solid. You need not find the volume. 2. Compute by switching the order of integration: dyd.x 3. Write the following integral in polar coordinates, then solve. arctan ( dyda ", 1.
1. For the solid bounded by the panes 2 = 1 - x and z=1-y in the first octant, one triple integral that describes the volume of the solid. The region is bounded by the x-axis and the curve y = √(2x-x^2), which is the top half of a circle centered at (1,0) with radius 1.
One possible order of integration is:
∫0^1 ∫0^(1-x) ∫0^(1-y) dzdydx
This means we integrate over z first, then y, then x. Another order of integration could be:
∫0^1 ∫0^x ∫0^(1-x-y) dzdydx
Here we integrate over z first, then x, then y.
Another possible order of integration is:
∫0^1 ∫0^1-x ∫0^1-y dzdxdy
Here we integrate over z first, then x, then y. This order of integration can also be rewritten in polar coordinates as:
∫0^(π/4) ∫0^(secθ-1) ∫0^(cscθ-1) r dzdrdθ
2. Compute by switching the order of integration:
∫0^2 ∫0^√(2x-x^2) dydx
First, let's sketch the region of integration. The region is bounded by the x-axis and the curve y = √(2x-x^2), which is the top half of a circle centered at (1,0) with radius 1.
We can switch the order of integration to integrate over x first, then y:
∫0^1 ∫0^(2-2y^2) dxdy
To find the limits of integration for x, we set y = √(2x-x^2) and solve for x:
y^2 = 2x - x^2
x^2 - 2x + y^2 = 0
(x-1)^2 = 1 - y^2
x = 1 ± √(1-y^2)
Since the curve is the top half of the circle, we take the positive square root:
x = 1 + √(1-y^2)
So the limits of integration for x are 0 to 2-2y^2. Integrating with respect to x first gives:
∫0^1 ∫0^(2-2y^2) dxdy = ∫0^1 (2-2y^2)dy = 4/3
3. Write the following integral in polar coordinates, then solve:
arctan (dy/dx)
We can write dy/dx in terms of polar coordinates using the chain rule:
dy/dx = (dy/dr)(dr/dθ)(1/dx/dθ)
Using the relationships x = rcosθ and y = rsinθ, we have:
dx/dθ = -rsinθ
dy/dθ = rcosθ, So
dy/dx = (dy/dr)(dr/dθ)(1/dx/dθ) = (cosθ)/(sinθ) = cotθ
Therefore, the integral becomes:
∫arctan(cotθ) dθ
To solve this integral, we use the identity arctan(x) + arctan(1/x) = π/2 for x > 0:
arctan(cotθ) = π/2 - arctan(tanθ)
So the integral becomes:
∫(π/2 - arctan(tanθ)) dθ
Integrating, we get:
(π/2)θ - ln|cosθ| + C
Where C is the constant of integration.
1. To find three other orders of integration for the solid bounded by the planes z = 1 - x, z = 1 - y, x = 0, y = 0, and z = 0 in the first octant, we can rearrange the given triple integral, which is given as:
∫∫∫_D dz dy dx
Now, we can find three other orders of integration:
a) ∫∫∫_D dx dz dy
b) ∫∫∫_D dy dx dz
c) ∫∫∫_D dy dz dx
2. To compute the volume of the solid by switching the order of integration, we can rewrite the given integral ∫∫ dy dx as: ∫∫ dx dy
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Suppose 60 seventh-grade students were surveyed.
How many can be expected to say that bike riding is
their favorite hobby?
Please IM IN NEED OF HELP
thx
We need to make some assumptions. Let's assume that the survey allowed students to choose one favorite hobby and that bike riding was one of the options.
We also need to know the percentage of students who chose bike riding as their favorite hobby. If this information is not given, we cannot accurately estimate the number of students who would say that bike riding is their favorite hobby.
Suppose that 30% of the surveyed students chose bike riding as their favorite hobby. To find out how many students this represents, we can use the following formula:
Expected number of students = Percentage of students x Total number of students surveyed
Plugging in the values we have, we get:
Expected number of students who say bike riding is their favorite hobby = 0.30 x 60 = 18
Therefore, we can expect that approximately 18 of the 60 seventh-grade students surveyed would say that bike riding is their favorite hobby, based on the assumption that 30% of the students chose this option.
It's important to remember that this is just an estimate based on the information we have. The actual number may be different depending on the survey results.
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Use the greatest common factor and the distributive property to write an equivalent expression in factored form. type your expression in the box.
9d+6e (pls answer this as soon as possible this is a quiz)
To write the given expression in factored form using the greatest common factor and distributive property, we need to find the largest common factor of 9 and 6, which is 3. Then we can factor out 3 from both terms, giving us 3(3d+2e). Therefore, the equivalent expression in factored form is 3(3d+2e).
This expression is simplified and shows that 3 is a common factor of both terms. In 100 words, this process involves identifying the greatest common factor between the terms and then using the distributive property to factor it out. This simplifies the expression and allows for easier calculations in further operations.
It is important to always look for common factors and simplify expressions whenever possible.
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Five machines are cutting 1.25-foot long
metal sheets. The machines are being
calibrated to ensure that they are cutting
the accurate length. The previous batches
for each machine are shown in the table.
Select all of the statements that are valid
for the data.
Only the statement "One machine is considerably more unreliable than the rest." is valid for the data.
How to get the valid statementsTotal number of correct cuts = 42 + 55 + 13 + 24 + 17 = 151
Total number of cuts = 100 + 100 + 100 + 100 + 100 = 500
Percentage of correct cuts = (151/500) * 100 = 30.2%
This statement is not valid, as only 30.2% of the cuts are the correct length.
One machine is considerably more unreliable than the rest."
By examining the number of correct cuts for each machine, we can see that Machine 3 has only 13 correct cuts, while the other machines have more than 17. This statement is valid.
3. When a machine misses the correct length, it tends to cut too long."
We need to compare the number of cuts that are too long (1.26-1.27 feet) with those that are too short (1.23-1.24 feet) across all machines:
Total number of cuts too long = 4 + 2 + 3 + 6 + 4 = 19
Total number of cuts too short = 980 + 72 + 67 = 1119
This statement is not valid, as the machines tend to cut too short rather than too long.
4. "Machine 5 will cut every batch the correct length at least 92% of the time."
To check this statement, we need to find the percentage of correct cuts for Machine 5:
Percentage of correct cuts for Machine 5 = (17/100) * 100 = 17%
This statement is not valid, as Machine 5 only cuts the correct length 17% of the time, which is less than 92%.
only the statement "One machine is considerably more unreliable than the rest." is valid for the data.
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There are 160 customers at Harris Teeter. 48 of them are children.What percent of the customers at Harris Teeter are adults?
PLEASE I NEED EXPLANATION
The percent of the customers at Harris Teeter that are adults is 70%
Calculating the percentage of the customers that are adultsFrom the question, we have the following parameters that can be used in our computation:
Customers = 160
Children = 48
using the above as a guide, we have the following:
Adults = Customers - Children
substitute the known values in the above equation, so, we have the following representation
Adults = 160 - 48
So, we have
Adults = 112
Next, we have
Percentage = 112/160 * 100%
Evaluate
Percentage = 70%
Hence, the percentage is 70%
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The table shows the dimensions of four boxes.
Drag tiles to order the volumes of the boxes from least to greatest
The order of the volumes of the boxes from least to greatest is Box D, Box B, Box C, Box A. Therefore, the correct option is D.
To determine the order of the volumes of the boxes from least to greatest, we will first calculate the volume of each box using the formula:
Volume = Length × Width × Height.
Hence,
1. Box A: Volume = 2in × 4.5in × 6in = 54 cubic inches
2. Box B: Volume = 6in × 2.5in × 3in = 45 cubic inches
3. Box C: Volume = 5in × 4.5in × 2.25in = 50.625 cubic inches
4. Box D: Volume = 2.5in × 2.25in × 3in = 16.875 cubic inches
Now, arrange the volumes in ascending order:
Box D (16.875), Box B (45), Box C (50.625), Box A (54)
Thus, the correct answer is D: Box D, Box B, Box C, Box A.
Note: The question is incomplete. The complete question probably is: The table shows the dimensions of four boxes. Which is the order of the volumes of the boxes from least to greatest?
Length Width Height
Box A 2in; 4.5in; 6in
Box B 6in; 2.5in; 3in
Box C 5in; 4.5in; 2.25in
Box D 2.5in; 2.25in; 3in
A) Box A Box B, Box C, Box D B) Box A, Box C, Box B, Box D C) Box B, Box D, Box A, Box C D) Box D, Box B, Box C, Box A.
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Aser these 5 math questions for branliest and points
Kira paid $15. 60 for 19. 5 centimeters of wire.
Find the unit price in dollars per centimeter.
If necessary, round your answer to the nearest cent.
HELP PLEASE!!! ASAP!!!
The unit price in dollars per centimeter is 80 cents.
:: Total wire length = 19.5 cm
:: Total amount paid = $15.60
Per unit price = [ (total amount paid) / (total wire length) ]
Per unit price = (15.60 / 19.5) $/cm
Per unit price = 0.8 ($/cm)
And as we know, $1 = 100 cents,
So,
$0.8 = 0.8 x 100 cents = 80 cents.
Therefore, the unit price in dollars per centimeter is 80 cents.
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Suppose a homing pigeon is released on an island at point C, which is 9 mi directly out in the water from a point B on shore, Point B is 20 mi downshore from the pigeon's home loft at point A. Assume that a pigeon flying over water uses energy at a rate 1.29 times the rate over land. Toward what points downshore from A should the pigeon fly in order to minimize the total energy required to get to the home loft at A? Point S ismiles away from point A. (Type an integer or decimal rounded to three decimal places as needed.)
The pigeon should fly directly from point C to point B, then fly along the shoreline to a point 10.387 miles away from point A (rounded to three decimal places). This can be found using the principle of minimizing the total distance traveled, taking into account the different energy rates over water and land.
To minimize the total energy required for the homing pigeon to get to its home loft at Point A, we need to find the optimal point downshore, Point S, to fly to. Using the given information, we can set up a function for the total energy.
Let x be the distance from Point A to Point S. Then, the pigeon will fly x miles over land and the remaining distance, 20-x miles, downshore from Point B to Point S. The distance from Point C to Point S can be found using the Pythagorean theorem:
CS = sqrt((20-x)^2 + 9^2)
Since the pigeon uses energy at a rate 1.29 times over water compared to land, we can write the total energy function as:
E(x) = x + 1.29 * CS
Now we need to minimize this function. To do so, we can take the derivative of E(x) with respect to x and set it equal to zero:
dE(x)/dx = 0
By solving this equation for x, we will find the optimal distance downshore from Point A to Point S. Once you have the value of x, you can say that Point S is x miles away from Point A (rounded to three decimal places, as needed).
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If a side of a square is doubled and an adjacent side is diminished by 3, a rectangle is formed whose area is numerically greater than the area of the square by twice the original side of the square. Find the dimensions of the original square
The dimensions of the original square is 8 by 8.
Let x be the original side length of the square. The area of the square is x². When one side is doubled and the adjacent side is diminished by 3, the rectangle's dimensions become 2x and (x-3). The area of the rectangle is (2x)(x-3) = 2x² - 6x.
According to the problem, the area of the rectangle is greater than the area of the square by twice the original side of the square, which is 2x. So we can set up the equation:
2x² - 6x = x² + 2x
Now, solve for x:
2x² - x² = 6x + 2x
x² = 8x
x = 8
So the dimensions of the original square are 8 by 8.
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A local deli sells 6-inch sub sandwiches for $2.95. Now the deli has decided to sell a “family sub” that is 50 inches long. If they want to make the larger sub price comparable to the price of the smaller sub, how much should it charge? Show all work.
Deli should charge $24.50 for the 50-inch family sub.
How much should the deli charge for a 50-inch?In a transaction, the price of something refers to amount of money that you have to pay in order to buy it. To make the prices comparable, we can use the unit price which is as follows>
The price per inch of 6-inch sub is:
= $2.95 / 6 inches
= $0.49/inch
To make 50-inch sub price, we will solve as:
= $0.49/inch * 50 inches
= $24.50
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For 2,000 paitents, blood-clotting time was normally distributed with a mean of 8 seconds and a standard deviation of 3 seconds. What percent had blood-clotting times between 5 and 11 seconds?
F. 69%
G. 34%
H. 49.5%
J. 47.5%
Thus, the percentage of the 2,000 paitents that had blood-clotting times between 5 and 11 seconds is 68.27% = 69%.
Explain about the normal distribution:The majority of data points in a continuous probability distribution called a "normal distribution" cluster around the range's middle point, while the ones that remain taper symmetrically towards either extreme. The distribution's mean is another name for the centre of the range.
Given data:
mean time μ = 8 secstandard deviation σ = 3 seconds5 < x < 11Then,
percent p (5 < x < 11 ) = z [(5 - μ) /σ < x < (11 - μ )/ σ]
p (5 < x < 11 ) = z [(5 - 8) /3 < x < (11 - 8 )/ 3]
p (5 < x < 11 ) = z [-1 < x < 1]
p (5 < x < 11 ) = z [0.8413 - 0.1586]
p (5 < x < 11 ) = 0.6827
p (5 < x < 11 ) = 68.27%
Thus, the percentage of the 2,000 paitents that had blood-clotting times between 5 and 11 seconds is 68.27% = 69%.
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What is 216 / 31? I keep on getting a decimal.
Answer:
6 30/31 as fraction and with Long Division its 6 with 30 Remainder
The tables represent hat sizes measured in inches for two softball teams.
Pelicans
21.5 25 22
21 22 23
22.5 24 21.5
22 23.5 22
23.5 22 24.5
Falcons
22.5 20 23.5
21 24 22
20.5 21.5 23
23 22.5 21
24 22 24.5
Which team has the largest overall size hat for their playersThe tables represent hat sizes measured in inches for two softball teams.
Pelicans
21.5 25 22
21 22 23
22.5 24 21.5
22 23.5 22
23.5 22 24.5
Falcons
22.5 20 23.5
21 24 22
20.5 21.5 23
23 22.5 21
24 22 24.5
Which team has the largest overall size hat for their players? Determine the best measure of center to compare and explain your answer.
Falcons; they have a larger median value of 22.5 inches
Pelicans; they have a larger median value of 22 inches
Falcons; they have a larger mean value of about 22 inches
Pelicans; they have a larger mean value of about 23 inches? Determine the best measure of center to compare and explain your answer.
Falcons; they have a larger median value of 22.5 inches
Pelicans; they have a larger median value of 22 inches
Falcons; they have a larger mean value of about 22 inches
Pelicans; they have a larger mean value of about 23 inches
Help me pls this is my last try
2. What is the smallest positive degree angle measure equivalent to tan-¹ (0.724)?
42.2°
31.0°
44.6°
35.9°
In a binomial experiment consisting of five trials, the number of different values that x (the number of successes) can assume is _____
In a binomial experiment consisting of five trials, the number of different values that x (the number of successes) can assume is 6.
A binomial experiment is a statistical experiment that meets four specific conditions: there are a fixed number of trials, each trial is independent of one another, there are only two possible outcomes (success or failure) in each trial, and the probability of success remains constant throughout the trials.
In this case, the binomial experiment consists of five trials, so the possible outcomes for x (the number of successes) can range from 0 successes to all 5 successes. To find the number of different values x can assume, simply add 1 to the total number of trials, as it includes the case of 0 successes.
Therefore, x can take on the following values: 0, 1, 2, 3, 4, or 5. As there are 6 possible values for x, the number of different values that x can assume in a binomial experiment consisting of five trials is 6.
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To begin a bacteria study, a petri dish had 2700 bacteria cells. Each hour since, the number of cells has increased by 5. 2%.
Let t be the number of hours since the start of the study. Let y be the number of bacteria cells.
Write an exponential function showing the relationship between y and t.
The exponential function y = [tex]2700(1.02)^t[/tex] models the growth of bacteria cells in a petri dish over time, with an initial population of 2700 cells and a growth rate of 2% per hour.
Exponential functions are often used to model situations where the growth or decay of a quantity depends on a constant proportionality factor.
In this case, the proportionality factor is the growth rate, which is represented by the constant 0.02 in the function. The factor (1 + r) represents the growth factor, which is the multiplier for the initial population to calculate the population after t hours. The larger the growth rate, the faster the population will grow, and the steeper the graph of the exponential function will be.
The equation y = [tex]2700(1.02)^t[/tex] can be used to make predictions about the growth of the bacteria population over time. For example, after one hour, the number of bacteria cells would be y = [tex]2700(1.02)^1[/tex] = 2754 cells. After two hours, the number of cells would be y = [tex]2700(1.02)^2[/tex] = 2812 cells, and so on.
It's worth noting that exponential growth cannot continue indefinitely, as there are always limiting factors that will eventually constrain the growth of a population. In the case of bacteria, the petri dish may eventually become overcrowded or run out of nutrients, which will slow or stop the growth of the bacteria population. Therefore, the exponential function y = [tex]2700(1.02)^t[/tex] is a model that is only valid for a certain range of values of t, beyond which other factors may come into play.
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Alec bought a house. By the end of the first year, the value of Alec's house had increased by 1%. By the end of the second year, its value had decreased by 10% of its value at the end of the first year. Use multipliers to work out the overall percentage decrease in the value of Alec's house for this two-year period
The overall percentage decrease in the value of Alec's house for the two-year period is 9.9%.
Let's assume the original value of Alec's house as $100. After the first year, the value of the house increased by 1%, which means the new value of the house is $101.
Now, in the second year, the value of the house decreased by 10% of its value at the end of the first year. Therefore, the new value of the house at the end of the second year can be calculated as $101 - (10/100)*$101 = $90.90.
To find the overall percentage decrease in the value of Alec's house for the two-year period, we can use the formula:
Overall percentage decrease = [(Original value - Final value)/Original value] * 100%
Substituting the values, we get,
Overall percentage decrease = [($100 - $90.90)/$100] * 100% = 9.9%
Therefore, the overall percentage decrease in the value of Alec's house for the two-year period is 9.9%.
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8-73.
For each diagram below, write and solve an equation for x
Answer:
a.x=100 equation: 540=(125·2)+90+(2x)
b. x=3 equation: 6x+18=2x+30
Step-by-step explanation:
a. We know that the interior angles of a pentagon have to equal 540 degrees.
So:
540=125+125+90 (hence the square) + x + x
simplify:
540=340+2x
simplify:
200=2x
x=100
b. Using the alternate exterior angles theorem we can say:
6x+18=2x+30
simplify:
4x+18=30
4x=12
x=3
Write the equation in standard form for the circle with center (8,0) and radius 3/3.
The equation in standard form for the circle with center (8,0) and radius 3/3 is (x - 8)² + y² = 1
To write the equation in standard form for the circle with center (8,0) and radius 3/3, we can use the following formula for a circle in standard form:
(x - h)² + (y - k)² = r²
Where (h, k) is the center of the circle and r is the radius. In this case, the center is (8,0) and the radius is 3/3, which simplifies to 1. Now, we can substitute the values of h, k, and r into the equation:
(x - 8)² + (y - 0)² = 1²
Since (y - 0) is just y, we can simplify the equation to:
(x - 8)² + y² = 1
So, the equation in standard form for the circle with center (8,0) and radius 3/3 is:
(x - 8)² + y² = 1
In summary, we used the standard form equation for a circle, substituted the given values for the center and radius, and simplified the equation to obtain the final answer.
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Rewrite the polynomial 2x^2+x^3+-7x+1 in standard form. Show your steps
So the polynomial 2x² + x³ - 7x + 1 in standard form is x³ + 2x² - 7x + 1.
What is the polynomial?To rewrite the polynomial 2x² + x³ - 7x + 1 in standard form, we need to write the terms in descending order of degree.
So we start with the highest degree term:
x³
Then we add the next highest degree term: 2x²
Followed by the next highest degree term: -7x
Finally, we add the constant term: +1
Putting all the terms together, we get:
x³ + 2x² - 7x + 1
So the polynomial 2x² + x³ - 7x + 1 in standard form is x³ + 2x² - 7x + 1.
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Marsha threw her math book off a 30 foot building. The equation of the book can be represented by the equation h=-16[tex]x^{2}[/tex]+24x+30. What is the maximum height
of Marsha's math book?
The maximum height of Marsha's math book is 36 feet.
To find the maximum height of Marsha's math book, we need to find the vertex of the parabolic equation h = [tex]-16x^2 + 24x + 30[/tex]. The vertex of a parabola is the highest or lowest point on the curve, depending on whether the parabola opens upward or downward.
To find the x-coordinate of the vertex, we can use the formula x = -b/2a, where a, b, and c are the coefficients of the quadratic equation [tex]ax^2 + bx + c[/tex]. In this case, a = -16 and b = 24, so we have:
x = -b/2a = -24/(2*(-16)) = 0.75
To find the y-coordinate of the vertex, we can substitute x = 0.75 into the equation h = [tex]-16x^2 + 24x + 30[/tex], which gives us:
h = [tex]-16(0.75)^2 + 24(0.75) + 30 = 36[/tex]
Therefore, the maximum height of Marsha's math book is 36 feet.
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gardens a square landscape plan is composed of three indoor gardens and one walkway that are all congruent. the gardens are centered around a square lounging area. if each side of the lounging area is 15 feet long, what is the area of one of the gardens?gardens a square landscape plan is composed of three indoor gardens and one walkway that are all congruent. the gardens are centered around a square lounging area. if each side of the lounging area is 15 feet long, what is the area of one of the gardens?
The area of one garden using each side of the lounging area is 15 feet long is equal to 56.25 square feet.
Shape of the garden landscape is square.
If the lounging area is a square with sides of length 15 feet,
Area of lounging area
= (15 feet) × (15 feet)
= 225 square feet
Four congruent sections of the landscape plan .
Three indoor gardens and one walkway.
Divide the lounging area into four equal square sections.
Each of the congruent sections has an area equal to,
Area of lounging area = 4 × area of one garden
Let's call the area of one garden be x.
⇒225 = 4x
Solving for x, we divide both sides by 4
⇒x = 225/4
⇒x = 56.25 square feet
Therefore, the area of one garden is 56.25 square feet.
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5. Daniel arrives at his campsite out of breath from his swim and jog. His sister tells him that he should have swam to the boat ramp that is only 200 feet from the campsite and then jogged. She claims that he would have arrived quicker this way. Is Daniel's sister correct? Support your answer mathematically
Since 2 is greater than 1, it would have been faster for Daniel to swim directly to the campsite. Therefore, Daniel's sister is incorrect.
We can solve this problem using the formula for distance, rate, and time, which is:
distance = rate x time
Let's assume that Daniel swims at a rate of s feet per second and jogs at a rate of j feet per second. If he swims directly to the campsite, the distance he needs to cover is the distance d between the campsite and the boat ramp, which is 200 feet. If he swims to the boat ramp and then jogs to the campsite, he will need to cover the distance d twice, once while swimming and once while jogging. The total distance he will cover is 2d = 400 feet.
If Daniel swims directly to the campsite, the time it will take him to cover the distance is:
time1 = d/s
If he swims to the boat ramp and then jogs to the campsite, the time it will take him to cover the distance is:
time2 = d/s + d/j
To compare the two times, we can take their ratio:
time2/time1 = (d/s + d/j)/(d/s) = 1 + j/s
If this ratio is less than 1, then Daniel's sister is correct, and he would have arrived quicker by swimming to the boat ramp and then jogging. If the ratio is greater than 1, then it would have been faster for him to swim directly to the campsite.
Substituting d = 200, we get:
time2/time1 = 1 + j/s = 1 + (j/s)*(200/200) = 1 + (200j)/(ds)
Since Daniel arrives at the campsite out of breath from his swim and jog, we can assume that his rates of swimming and jogging are roughly equal. Let's assume s = j = r, where r is the common rate of swimming and jogging. Substituting this into the ratio, we get:
time2/time1 = 1 + (200r)/(dr) = 1 + 200/d
To determine if Daniel's sister is correct, we need to compare this ratio to 1. If time2/time1 is less than 1, then Daniel's sister is correct. If it is greater than 1, then it would have been faster for Daniel to swim directly to the campsite.
Substituting d = 200, we get:
time2/time1 = 1 + 200/200 = 2
Since 2 is greater than 1, it would have been faster for Daniel to swim directly to the campsite. Therefore, Daniel's sister is incorrect.
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The diameter of the base of a cone is 8 inches and the height is twice the radius. What is the volume of the cone? Use 3.14 for π
.
Group of answer choices
133.97 in3
401.92 in3
50.24 in3
66.99 in3
The volume of the cone is approximately 133.97 cubic inches. So, correct option is A.
The diameter of the base of a cone is 8 inches, which means that the radius is 4 inches (since radius = diameter/2). The height of the cone is twice the radius, which means the height is 2 x 4 = 8 inches.
The formula for the volume of a cone is V = (1/3)πr²h, where r is the radius and h is the height.
Substituting the values of r and h into the formula, we get:
V = (1/3)π(4²)(8)
V = (1/3)π(16)(8)
V = (1/3)π(128)
V ≈ 133.97 in³
Therefore, correct option is A.
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On a standard dice, the sum of the numbers of dots on opposite faces is always 7. Four standard dice are glued together, as shown. What is the minimum number of dots that could lie on the whole surface?
The minimum number of dots that could lie on the whole surface of four standard dice glued together is 56.
This can be achieved by placing the faces with 1 dot opposite the faces with 6 dots, the faces with 2 dots opposite the faces with 5 dots, and the faces with 3 dots opposite the faces with 4 dots on all four dice. Since the sum of the numbers on opposite faces of each individual die is always 7, the sum of the numbers on opposite faces of the four glued-together dice is also always 7. Therefore, the minimum number of dots on the whole surface is 7 times the number of faces, which is 7 x 8 = 56.
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N Tools
Find the experimental probability that only 1 of
4 children in a family is a girl.
The problem has been simulated by tossing
coins (one to represent each child). Let "heads"
represent a boy and "tails" represent a girl. A
sample of 20 coin tosses is shown.
HTHн
HTTH
TTTT
THTT
НТНТ
HHTT
HHHT
THHT
HTTH
TTHH
HTTT
НТНТ
TTHH
ТНТН
HTHH
ТЕНТ
HTTT
НТНТ
HHHT
HHHH
Experimental Probability = [?]%
Enter
45% is the experimental probability that only 1 of 4 children in a family is a girl.
To find the experimental probability that only 1 of 4 children in a family is a girl, we need to count the number of times this outcome occurs in the sample and divide it by the total number of outcomes. In this case, we have 20 coin tosses, and we are looking for sequences with exactly 1 "tails" (girl) and 3 "heads" (boys).
Here are the sequences with exactly 1 girl:
HTHH
HHTT
HHHT
THHT
HTTH
TTHH
THTT
TTHH
THTT
There are 9 such sequences out of the total 20 coin tosses. Therefore, the experimental probability is:
(9/20) * 100% = 45%
The experimental probability that only 1 of 4 children in a family is a girl is 45%.
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