The size of the square that needs to be cut out of each corner to create a box of maximum volume is 5/3 inches.
To determine the function f given that f'(x)=6x² – 5 sin x+eˣ and f(0) = 20, we need to integrate f'(x) with respect to x to obtain f(x), and then use the initial condition f(0) = 20 to find the value of the constant of integration.
Integrating f'(x) with respect to x, we have:
f(x) = 2x³ + 5 cos x + eˣ + C
where C is the constant of integration.
Using the initial condition f(0) = 20, we have:
f(0) = 2(0)³ + 5 cos 0 + e⁰ + C = 6 + C = 20
Therefore, the constant of integration is C = 14, and the function f(x) is:
f(x) = 2x³ + 5 cos x + eˣ + 14
To determine the size of the square that needs to be cut out of each corner of a standard piece of notebook paper to create a box of maximum volume, we can start by drawing a diagram of the box and labeling the sides as follows:
| |
| |
| | h
| |
|__________|
L
Let x be the length of each side of the square that is cut out of each corner. Then, the length and width of the base of the box will be L - 2x and 11 - 2x, respectively, and the height of the box will be x. Therefore, the volume V of the box can be expressed as:
V(x) = x(L - 2x)(11 - 2x)
Expanding and simplifying, we get:
V(x) = -4x³ + 46x² - 110x
To find the size of the square that maximizes the volume of the box, we need to find the value of x that maximizes V(x). This can be done by finding the critical points of V(x) and determining whether they correspond to a maximum or minimum.
Taking the derivative of V(x) with respect to x, we get:
V'(x) = -12x² + 92x - 110
Setting V'(x) = 0 and solving for x, we get:
x = 5/3 or x = 11/6
To determine whether these values correspond to a maximum or minimum, we can use the second derivative test. Taking the second derivative of V(x) with respect to x, we get:
V''(x) = -24x + 92
Evaluating V''(5/3) and V''(11/6), we find that:
V''(5/3) = -4 < 0, so x = 5/3 corresponds to a maximum.
V''(11/6) = 20 > 0, so x = 11/6 corresponds to a minimum.
Therefore, the size of the square that needs to be cut out of each corner to create a box of maximum volume is 5/3 inches.
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The function models the amount of money a new business has made over x months. Negative values represent debt. Over what interval is the amount increasing?
If the line has a positive slope, it means that the amount is increasing, and if it has a negative slope, it means that the amount is decreasing.
Based on the given information, the function models the amount of money a new business has made over x months. Therefore, we can assume that the function is a linear function, and its graph is a straight line.
Given that negative values represent debt, we can assume that the slope of the line is positive over the interval where the function takes positive values. Therefore, the interval over which the amount is increasing is from zero to infinity, which means that the business is making money and not in debt.
It's essential to note that if the function has a negative slope over any interval, then the amount is decreasing, and the business is losing money. However, we cannot determine this information from the given function's information.
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The school swim team swam a combined distance of 346. 725 meters during swim practice. Using number names, how would you say 346. 725? (6. NBT. 3_a. ) *
The number name of 346. 725 is three hundred forty-six point seven two five.
A number name refers to the name that is used to describe the number in words. This is helpful in communicating it orally.
To convert the given number to word form, we do as follows:
1. We check the digits of the number before decimals
In this case, the number of digits in the question is 3
2. The highest place value is then checked.
It comes out to be hundreds
3. We name it accordingly and add a suffix to the face value
This is 3 and the name comes out to be Three hundred
4. We continue it till we encounter the decimal
We get the number as Three hundred forty-six
5. Then we mention the word decimal or point
The result is Three hundred forty-six point
6. The number after the decimal is written as it is
Hence, the name comes out to be Three hundred forty-six point seven two five.
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The target to the right is in your backyard. What is the probability of hitting the bulls eye (center circle) when you shoot an arrow? The radius of the bulls eye is 2ft and the radius of the target is 6ft. (Use 3. 14 for pi)
The probability of hitting the bulls eye (center circle) when we shoot an arrow with radius of the bulls eye is 2ft and the radius of the target is 6ft is 11.1%.
Assuming that we are shooting randomly at the target and the arrow can land anywhere within the target area, the probability of hitting the bull's eye (center circle) can be calculated as the ratio of the area of the bull's eye to the area of the entire target.
The area of the bull's eye can be calculated as follows:
Area of bull's eye = π x (radius of bull's eye)²
Area of bull's eye = 3.14 x 2²
Area of bull's eye = 12.56 square feet
The area of the entire target can be calculated as follows:
Area of target = π x (radius of target)²
Area of target = 3.14 x 6²
Area of target = 113.04 square feet
Therefore, the probability of hitting the bull's eye can be calculated as:
Probability of hitting bull's eye = Area of bull's eye / Area of target
Probability of hitting bulls eye = 12.56 / 113.04
Probability of hitting bulls eye = 0.111 or approximately 11.1%
So, the probability of hitting the bull's eye (center circle) when we shoot an arrow randomly at the target is approximately 11.1%.
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Help pls I need help with sand and a word with my
Answer: X= 4/5x + 8
Step-by-step explanation: Distribution factor
Forests are complex, evolving ecosystems. For
instance, pioneer tree species can be displaced by successional species better adapted
to the changing environment. Ecologists mapped a large Canadian forest plot
dominated by Douglas fir with an understory of western hemlock and western red cedar.
Sapling trees (young trees shorter than 1.3 meters) are indicative of the future of a
forest. The 246 sapling trees recorded in this sample forest plot were of the following
types:
Are there equal proportions of RC, DF, and WH among sapling trees in this forest?
1. State your null and alternative hypothesis. (10 points)
2. Report appropriate test statistics and p-value. (20 points)
3. With alpha=0.05, state your conclusion. (10 points)
4. Investigate components of your test statistics. What does your analysis suggest
about this forest’s successional stage? (10 points)
P(Dead l WH) = 0.23/0.48 = 0.4792
Here the western red cedar is relatively young.
How to solve(a) P(Dead l RC) = 0.02/0.20 = 0.10
P(Dead l DF) = 0.16/0.32 = 0.50
P(Dead l WH) = 0.23/0.48 = 0.4792
Here the western red cedar is relatively young.
(b) P(sapling l RC) = 0.08/0.20 = 0.4
P(sapling l DF) = 0.00/0.32 = 0
P(sapling l WH) = 0.04/0.48 = 0.08333
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can someone help me please
The circumference of a circle is the distance around the edge or perimeter of the circle.
What is the circumference of a circle?1) Radius = 5cm/2 = 2.5 cm
2) Radius = 28 mm/2 = 14 mm
3) Radius = 3 1/2 m * 1/2 = 3/4 m
4) diameter = 2 * 6cm = 12 cm
5) diameter = 2 * 2m = 4 m
6) diameter = 2 * 0.8 ft = 1.6 ft
Circumference can be calculated using the formula:
C = 2πr
7) circumference = 2πr = 2 * 3.14 * 10/2 = 31.4 in
8) circumference = 2 * 7 * 3.14 = 43.96 in
9) Circumference = 2 * 3.14 * 18/2 = 56.52 in
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50 PONTS Triangle LMN has vertices at L(−1, 4), M(−1, 0), and N(−3, 4) Determine the vertices of image L′M′N′ if the preimage is rotated 90° clockwise about the origin.
L′(4, 1), M′(0, 1), N′(4, 3)
L′(−1, −4), M′(−1, 0), N′(−3, −4)
L′(−4, −1), M′(0, −1), N′(−4, −3)
L′(1, −4), M′(1, 0), N′(3, −4)
The coordinates of the resulting triangle are L'(4, 1), M'(0, 1), and N'(4, 3)
What are the coordinates of the resulting triangle?From the question, we have the following parameters that can be used in our computation:
Triangle LMN has vertices at L(−1, 4), M(−1, 0), and N(−3, 4
This means that
L(−1, 4), M(−1, 0), and N(−3, 4Rotation rule = 90° clockwise around the origin.The rotation rule of 90° clockwise around the origin is
(x,y) becomes (y,-x)
So, we have
Image = (y, -x)
Substitute the known values in the above equation, so, we have the following representation
L'(4, 1), M'(0, 1), and N'(4, 3)
Hence, the coordinates of the resulting points, are L'(4, 1), M'(0, 1), and N'(4, 3)
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Answer:L′(4, 1), M′(0, 1), N′(4, 3)
Step-by-step explanation:
I am in the middle of taking the quiz and I believe this is the correct answer!
Which expression are equivalent to the given expression?
By the rule of indices the given expression is equivalent to x⁻² y⁻⁵.(option B)
What is index?
In mathematics, Index in plural indices is the power or exponent which is raised to a number or a variable. For example, for the number 2⁵, 5 is the index of 2. Which gives the value 32.
Given expression is y⁻⁸ y³ x⁰ x⁻²
Here we use the law of indices.
By the law of indices it can be stated that the indices at the time of multiplication is being added up. That is zᵃ × zᵇ = zᵃ⁺ᵇ.
Using this concept of indices the given expression can be rewritten as
y⁻⁸⁺³ x⁰⁻²
= y ⁻⁵ x⁻²
= x⁻² y⁻⁵
Hence, the given expression is equivalent to x⁻² y⁻⁵.
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I have no congruent sides. One of my angles has a measure of 100 degrees. Answer with drawing of the triangle
I am a(n and triangle
You are an scalene triangle.
How can you identify the type of triangle when given the information that it has no congruent sides and one angle measuring 100 degrees?You are a scalene triangle.
A scalene triangle is a type of triangle where all three sides have different lengths, and no two angles are congruent. In this case, you mentioned that one of the angles has a measure of 100 degrees.
Here's a simple diagram of a scalene triangle to help illustrate:
\
\
\
\
\
\
In the diagram, the angles are not drawn to scale, but it represents a scalene triangle where one angle measures 100 degrees. The sides of the triangle would have different lengths, distinguishing it from an equilateral or isosceles triangle where at least two sides are congruent.
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Mr. vara is designing post caps in the shape of a square pyramid for a fence that he just built in his backyard. the caps are solid wood and each one has a volume of 94.5 cubic centimeters.
If Mr. Vara's square pyramid post caps have a volume of 94.5 cubic centimeters, and he assumes that the height and side length are equal, then the side length of each cap should be approximately 6.04 centimeters.
Mr. Vara is designing post caps in the shape of a square pyramid for his fence. The caps are solid wood and have a volume of 94.5 cubic centimeters. To calculate the dimensions of the pyramid, we need to use the formula for the volume of a square pyramid, which is V = (1/3)bh, where b is the area of the base and h is the height of the pyramid.
Since the caps are square pyramids, the base is a square. Let's call the side length of the base s. Then the area of the base is s². We can rearrange the formula for volume to get h = (3V)/b. Plugging in the given volume of 94.5 cubic centimeters, we get:
h = (3 x 94.5) / s²
h = 283.5 / s²
We still need to find the side length s. We can use the fact that the volume of the pyramid is also equal to [tex](1/3)s^{2h[/tex]. Plugging in the volume and height from above, we get:
94.5 = (1/3)s²(283.5 / s²)
94.5 = 94.5
This equation simplifies to 1 = 1, which doesn't give us any information about s. However, we can make an assumption about the dimensions of the pyramid. Let's say that the height h is equal to the side length s. Then we can solve for s using the volume formula:
[tex]94.5 = (1/3)s^{2s[/tex]
94.5 = (1/3)s³
283.5 = s³
s = 6.04
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Want is the measure of
4. Question: How can an antiderivative of a velocity function be used to find displacement of the body? Question: If velocity is negative, how does this impact the problem of finding displacement? 5.
If velocity is negative,The body is moving backwards instead of forwards, so the displacement will be measured as a negative value.
An antiderivative of a velocity function can be used to find displacement of the body by integrating the velocity function over a given time interval. The result of this integration will give the total displacement of the body over that time interval.
If the velocity is negative, it means that the body is moving in the opposite direction to the positive direction of the coordinate system. This has an impact on the problem of finding displacement because the displacement will also be negative. In other words, the body is moving backwards instead of forwards, so the displacement will be measured as a negative value.
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Write the absolute value inequality in form |x-b| c that has the solution set x<-5 or x>7
The absolute value inequality in form |x-b| < c that has the solution set x<-5 or x>7 is |x-1| < 7
To write an absolute value inequality in the form |x-b| < c, we need to think about what this form means.
The expression |x-b| represents the distance between x and b on the number line. Therefore, |x-b| < c means that the distance between x and b is less than c.
Now, let's consider the given solution set: x < -5 or x > 7. We can see that the midpoint between -5 and 7 is 1, so we choose b = 1. Then, we need to determine c, which is the maximum distance between b and any of the solutions.
If we take x = -6 (which is less than -5), then |x-b| = |-6-1| = 7. Similarly, if we take x = 8 (which is greater than 7), then |x-b| = |8-1| = 7. Therefore, the maximum distance is 7.
Putting it all together, we get the absolute value inequality:
|x-1| < 7
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find the exact area of a square with a diagonal of 8 inches
Answer:
(8/√2)^2 = 64/2 = 32 square inches
In a recent report, Joe's, a Memphis-style barbecue chain, states that 11% of its customers order for delivery. A random sample of 6 Joe's customers is chosen. Find the probability that at most 1 of them order for delivery. " Do not round your intermediate computations, and round your answer to three decimal places.
P(X ≤ 1) = P(X = 0) + P(X = 1) ≈ 0.901
So the probability that at most 1 customer in the sample orders for delivery is approximately 0.901, rounded to three decimal places.
To solve this problem, we can use the binomial distribution since we have a fixed number of trials (6) and each trial can result in one of two outcomes (ordering for delivery or not).
Let X be the number of customers in the sample who order for delivery. Then X follows a binomial distribution with parameters n = 6 and p = 0.11 (the probability of ordering for delivery).
We want to find the probability that at most 1 customer orders for delivery. This can be written as:
P(X ≤ 1) = P(X = 0) + P(X = 1)
To calculate these probabilities, we can use the binomial probability formula:
P(X = k) = (n choose k) *[tex]p^k[/tex]*[tex](1 - p)^(n - k)[/tex]
where (n choose k) is the binomial coefficient, which represents the number of ways to choose k items from a set of n distinct items.
Using this formula, we can calculate:
P(X = 0) = (6 choose 0) * 0.11^0 * [tex]0.89^6[/tex] ≈ 0.530
P(X = 1) = (6 choose 1) * 0.11^1 * 0.89^5 ≈ 0.371
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you select a marble without looking and then put it back if you do this 90 times what is the best predictions that you will pick an orange or a pink marble?
If you select a marble without looking and then put it back if you do this 90 times, the probability of selecting an orange or a pink marble on any given trial is 1/2 or 0.5.
If you select a marble without looking and then put it back, the probability of selecting any particular marble on any given trial is the same for all marbles. Assuming there are only two possible outcomes - selecting an orange marble or a pink marble - the probability of selecting an orange or a pink marble on any given trial is 1/2 or 0.5.
Since the probability of selecting an orange or a pink marble on each trial is the same, the best prediction for the number of times you will select an orange or a pink marble out of 90 trials is an equal number of times for each color. Therefore, the best prediction for the number of times you will select an orange or a pink marble is 45 times each.
This is only a prediction based on probability, and the actual number of times you select an orange or a pink marble may differ from the prediction due to chance. However, over a large number of trials, the actual outcomes should approach the predicted probabilities.
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A newspaper for a large city launches a new advertising campaign focusing on the number of digital subscriptions. the equation s(t)=31,500(1.034)t approximates the number of digital subscriptions s as a function of t months after the launch of the advertising campaign. determine the statements that interpret the parameters of the function s(t).
"t" increases, and the number of digital subscriptions grows exponentially at a rate of 3.4% per month.
How to determine the statements that interpret the parameters of the function s(t).The function s(t) = 31,500(1.034)^t gives an approximation of the number of digital subscriptions s as a function of t months after the launch of the advertising campaign.
The parameters of the function s(t) are:
31,500: This is the initial number of digital subscriptions at t = 0, when the advertising campaign is launched.
1.034: This is the growth rate of the number of digital subscriptions per month. It represents the percentage increase in the number of subscriptions each month due to the advertising campaign. Specifically, each month the number of subscriptions is multiplied by 1.034, which is the same as increasing it by 3.4%.
t: This is the time in months after the launch of the advertising campaign. It is the independent variable of the function that determines the number of digital subscriptions at any given time t.
Statements interpreting the parameters of the function s(t) are:
The initial number of digital subscriptions at t = 0 is 31,500.
For every month after the launch of the advertising campaign, the number of digital subscriptions increases by 3.4%, or a factor of 1.034.
The parameter t represents the time in months after the launch of the advertising campaign. As t increases, the number of digital subscriptions grows exponentially at a rate of 3.4% per month.
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An electrical charge of 0. 000003 coulombs is 0. 06 m away from another electrical charge of 0. 0000009 coulombs. Solve this equation
The force between the two charges is 1.69 x[tex]10^-^1^5[/tex] N
How to find the equation?The force between two electric charges can be calculated using Coulomb's law, which states that the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. Mathematically, it can be expressed as:
F = k * (q1 * q2) / d²
where F is the force, k is Coulomb's constant (9 x 10⁹ N m² / C²), q1 and q2 are the magnitudes of the two charges, and d is the distance between them.
Substituting the given values, we get:
F = (9 x [tex]10^9[/tex] N m² / C²) * (0.000003 C) * (0.0000009 C) / (0.06 m)²
Simplifying this expression gives:
F = 1.69 x[tex]10^-^1^5[/tex] N
Therefore, the force between the two charges is 1.69 x [tex]10^-^1^5[/tex] N.
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Use the appropriate compound interest formula to find the amount that will be in each account, given the stated coins, $47.000 invested at 5% annual interest for 5 years compounded (a) annually: (b) semiannually
a)The amount in the account after 5 years, compounded annually, will be approximately $60,625.06.
b) The amount in the account after 5 years, compounded semiannually, will be approximately $61,588.88.
The amount in the account after 5 years can be calculated using the compound interest formula. For an initial investment of $47,000 at an annual interest rate of 5%, compounded annually or semiannually, the amount in the account will be different.
(a) For compounding annually:
The compound interest formula is given by:
A = P(1 + r/n)^(nt)
where:
A = the amount after time t
P = principal amount (initial investment) = $47,000
r = annual interest rate = 5% or 0.05 (as a decimal)
n = number of times interest is compounded per year = 1 (since it is compounded annually)
t = time period = 5 years
Plugging in the given values, we get:
A = 47000(1 + 0.05/1)^(1*5)
A = 47000(1.05)^5
A ≈ $60,625.06
So, the amount in the account after 5 years, compounded annually, will be approximately $60,625.06.
(b) For compounding semiannually:
The only difference from the above calculation is the value of 'n', which is the number of times interest is compounded per year. In this case, 'n' will be 2, as interest is compounded semiannually (twice a year).
Plugging in the given values, we get:
A = 47000(1 + 0.05/2)^(2*5)
A = 47000(1.025)^10
A ≈ $61,588.88
So, the amount in the account after 5 years, compounded semiannually, will be approximately $61,588.88.
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Llus
Find x.
Round to the nearest tenth:
31°
х
y
400 ft
x = [? ]ft
Pls help me
Using the sine function and the given information, we can find that length of y is approximately 203.3 feet.
We know that exterior angle of a triangle is equal to the sum of its opposite interior angles. So, we can find angle BAC as follows
BAC + ABC = 180 degrees
As sum of interior angles of a triangle
BAC + 90 = 180
BAC = 90 degrees
Now, we can use the sine ratio to find y
sin(31) = y/400
y = 400 * sin(31)
y ≈ 203.3 ft
Therefore, y is approximately 203.3 ft when rounded to the nearest tenth.
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--The given question is incomplete, the complete question is given
" Find y. Round to the nearest tenth:
In triangle ABC, 31° is the outer angle at A which is in between a line extended from A parallel to BC.
X is CA
у is AB
400 ft is BC
Angle B is right angle
y = [ ? ]ft Enter"--
HELP ME PLEASE I DON'T UNDERSTAND
Answer:
34/73
Step-by-step explanation:
37 + 34 + 2 = number of customers = 73
73 is our denominator.
34 is the number of people who used a credit card.
34 is our numerator.
Put the two together, and you get 73! Enjoy!
Each day three church bells are rung in a random order. what is the probability that the smallest bell rings first three days in a row?
The probability that the smallest bell rings the first three days in a row is 1/27. when Each day three church bells are rung in a random order
In the given data there are 3 bells in the church in which there is a small bell and the three bells are rung in a random order. we need to find the probability that the smallest bell rings the first three days in a row.
The probability that the smallest bell rings on any given first day can be given as = 1/3
Because there are three bells and each bell has an equal chance of being rung first. The probability that this happens three days in a row is given as
= (1/3) × (1/3) × (1/3)
= 1/27
Therefore, the probability that the smallest bell rings the first three days in a row is 1/27
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Someone please help with this i need it really fast
Answer:
67. 50
Step-by-step explanation:
15x4.50= 67.5
Here is a set of data showing the test scores for US History class:
56, 88, 70, 72, 90, 85, 99, 65, 66, 54, 74, 85, 91, 92, 72, 88, 97, 62, 88 Create a stem and leaf plot to show this data. Hint: Decide how many stems you will need.
can somebody help me ?
Hi! I'd be happy to help you create a stem and leaf plot using the provided set of data for US History class test scores.
Step 1: Arrange the data in ascending order.
54, 56, 62, 65, 66, 70, 72, 72, 74, 85, 85, 88, 88, 88, 90, 91, 92, 97, 99
Step 2: Determine the range of the data.
The data ranges from 50s to 90s, so we will need 5 stems: 5, 6, 7, 8, and 9.
Step 3: Create the stem and leaf plot using the stems and corresponding leaves (the units digits of the data).
5 | 4 6
6 | 2 5 6
7 | 0 2 2 4
8 | 5 5 8 8 8
9 | 0 1 2 7 9
Here is the completed stem and leaf plot for the US History class test scores. The stems represent the tens digits (50s, 60s, 70s, 80s, 90s), and the leaves represent the units digits of the scores in each range.
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you purchase a computer for 875 plus 5% sales tax. you decide to finance it through the stores 0% program for 12 months. the terms state that you must pay
The amount of interest paid for missing the payment in the eight month would be C. $87.28
How to find the interest ?First, find the total cost of the computer including the sales price :
= 875 x ( 1 + 0. 05 )
= 875 x 1. 05
= $ 918. 75
Then, find the monthly rate of interest :
= 14. 25 / 12
= 1. 1875 %
This leads to an eight month interest rate of :
= 8 x 1. 1875 %
= 9. 5 %
The interest to be paid:
= 9. 5 % x 918. 75
= $ 87. 28
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Full question is:
You purchase a computer for $875.00 plus 5% sales tax. You decide to finance it through the store's 0% program for 12 months. The terms state that you must pay $100.00/month and that if you miss a payment, you will be assessed a late fee of $39.00 plus the interest accrued to that point at a 14.25% APR. If you miss a payment in the eighth month, how much interest will you be charged?. A. $83.13. B. $16.63. C. $87.28. D. 58.19
450 cubic centimetres of wood is used to make a solid cylindrical ornament. the radius of the base of the ornament is 5 centimetres. what is the height of the cylindrical ornament? (formula = π x radius² x height)
a) 4.5 cm
b) 5.7 cm
c) 6.3 cm
d) 7.5 cm
The height of the cylindrical ornament is approximately 5.7 centimetres (option b).
To find the height of the cylindrical ornament given that 450 cubic centimetres of wood is used and the radius of the base is 5 centimetres, you can use the formula for the volume of a cylinder: V = π × radius² × height.
Step 1: Write down the given values.
Volume (V) = 450 cubic centimetres
Radius (r) = 5 centimetres
Step 2: Plug the given values into the formula.
450 = π × (5)² × height
Step 3: Solve for the height.
450 = π × 25 × height
450 = 78.54 × height
Step 4: Divide both sides by 78.54 to find the height.
height = 450 / 78.54
height ≈ 5.7 centimetres
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Y’all I’m so confused, I got 38792.3. What am I doing wrong?
Answer:
38,772.72
Step-by-step explanation:
The formula for a sphere is
V=4/3 pi r^3
=4/3 3.14 21^3
=4/3 3.14 9,261
=4/3 29,079.54
=38,772.72
[tex]\textit{volume of a sphere}\\\\ V=\cfrac{4\pi r^3}{3}~~ \begin{cases} r=radius\\[-0.5em] \hrulefill\\ r=42 \end{cases}\implies V=\cfrac{4\pi (42)^3}{3}\implies \stackrel{ using~\pi =3.14 }{V\approx 310181.76}[/tex]
69x 10x 6969x 8008x696969696969
What is the domain of the rational function f of x is equal to the quantity x squared plus x minus 6 end quantity over the quantity x cubed minus 3 times x squared minus 16 times x plus 48 end quantity question mark {x ∈ ℝ| x ≠ –4, –2, 3, 4} {x ∈ ℝ| x ≠ –4, 3, 4} {x ∈ ℝ| x ≠ –4, 4} {x ∈ ℝ| x ≠ –2, 3}
The domain of the rational function is: option (2) {x ∈ ℝ | x ≠ -4, 3, 4}
What is Rational number ?A rational number is any number that can be expressed as the ratio or fraction of two integers, where the denominator is not zero. In other words, a rational number is a number that can be written in the form of p/q, where p and q are integers, and q is not equal to zero.
The domain of a rational function is the set of all real numbers for which the function is defined, and the denominator is not equal to zero. So, we need to find the values of x for which the denominator of the given rational function is not zero.
The denominator of the given rational function is:
x³ - 3x² - 16x + 48
We can factor this polynomial using synthetic division or polynomial long division:
x³- 3x² - 16x + 48 = (x - 4)(x - 3)(x + 4)
So, the denominator of the rational function is not defined when:
x - 4 = 0 or x - 3 = 0 or x + 4 = 0
Solving these equations, we get:
x = 4 or x = 3 or x = -4
Therefore, the domain of the given rational function is:
{x ∈ ℝ| x ≠ –4, 3, 4}
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John was visiting four cities that form a rectangle on a coordinate grid at A(O.
4), B(4,1). C(3. 1) and D(-1. 2). If he visited all the cities in order and ended up
where he started. What is the distance he traveled? Round your answer to the
nearest tenth
If he visited all the cities in order A(O,4), B(4,1). C(3. 1) and D(-1. 2). then he traveled 12.3 units distance ( nearest tenth).
John visited four cities that form a rectangle on a coordinate grid at A(0, 4), B(4, 1), C(3, 1), and D(-1, 2). If he visited all the cities in order and ended up where he started, the distance he traveled can be found by calculating the perimeter of the rectangle.
Calculate the distance between consecutive points.
AB = √[(4-0)^2 + (1-4)^2] = √[16 + 9] = √25 = 5
BC = √[(3-4)^2 + (1-1)^2] = √[1 + 0] = √1 = 1
CD = √[(-1-3)^2 + (2-1)^2] = √[16 + 1] = √17 ≈ 4.1 (rounded to nearest tenth)
DA = √[(0-(-1))^2 + (4-2)^2] = √[1 + 4] = √5 ≈ 2.2 (rounded to nearest tenth)
Calculate the total distance traveled (perimeter of the rectangle).
Total Distance = AB + BC + CD + DA = 5 + 1 + 4.1 + 2.2 = 12.3
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