The area of the rectangle will grow at a rate of 100 square inches per second if the width of the rectangle is 10 inches.
let the width w of the rectangle is 1 inch and the length of the rectangle is "2w" inches.
The area of the rectangle is given by
A = length x width = (2w)(w) =[tex]2w^2[/tex] square inches if the width of the rectangle is 10 inches
To find out how fast the area of a rectangle is changing, we need to differentiate the area over time.
dA/dt =[tex]d/dt (2w^2)[/tex] = 4w(dw/dt)
where dw/dt is the rate of change in width.
as the length of the rectangle is increasing at a rate of 5 inches/second.
d(2w)/dt = 5 inches/second
Simplifying the above formula to:
2(dw/dt) = 5 inches/second
dw/dt = 5/2 inch/second
If the rectangle is 10 inches wide, then:
w = 10 inches
dw/dt = 5/2 inch/second
Substitute these values into the formula for dA/dt.
dA/dt = 4w(dw/dt) = 4(10)(5/2) = 100 square inches per second
therefore, if the width of the rectangle is 10 inches, the area of the rectangle will grow at a rate of 100 square inches per second.
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x + 2xyz
9x3y2
18x2 + 5ab − 6y
4x4 + 3x2 − x − 4
The expressions when factorized are x(1 + 2yz), 9x^3y^2 and 6(3x^2 − 2y) + (5ab + 6y)
Factorizing the expressionsFactorization of X + 2xyz:
We can factor out x from the given expression to get:
x(1 + 2yz)
Therefore, the factorization of X + 2xyz is x(1 + 2yz).
Factorization of 9x3y2:
The given expression, 9x3y2, is already in its simplest form and cannot be factorized further.
Factorization of 18x2 + 5ab − 6y:
We can use the grouping method to factorize the given expression:
(18x2 − 12y) + (5ab + 6y)
6(3x2 − 2y) + (5ab + 6y)
Therefore, the factorization of 18x2 + 5ab − 6y is 6(3x2 − 2y) + (5ab + 6y).
Factorization of 4x4 + 3x2 − x − 4:
Cannot be factorized
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7. Higher Order Thinking The manager of a restaurant wants to choose a different side dish for the special each week. She surveys customers about side dish preferences and records the results in the table shown at the right. How can she use 10 index cards to simulate choosing a side dish?
This method ensures that the side dish selection reflects customer preferences while incorporating an element of randomness.
Hi! To simulate choosing a side dish using 10 index cards, the manager can follow these steps:
1. Assign a number to each side dish based on the survey results.
2. Determine the probability of each side dish being chosen according to customer preferences.
3. Based on the probabilities, allocate the appropriate number of index cards to each side dish (e.g., if a dish has a 30% chance of being chosen, it would be assigned 3 out of the 10 index cards).
4. Write the side dish name or its assigned number on the allocated index cards.
5. Shuffle the index cards and randomly draw one card each week to determine the special's side dish.
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during the experiment, the researcher noticed that individual mice sometimes switched which forepaw they used to retrieve the food. so she repeated this study on the same 48 mice, but measured each one 10 times and she recorded which forepaw was used each time. what would be the problem with repeating the analysis ran in part (a) with her new dataset of 480 observations? be specific
a. Based on the chi-squared goodness of fit test, we do not have enough evidence to conclude that mice show a preference towards one paw over the other, with a calculated chi-squared value of 2.5 and a critical value of 3.84 at a significance level of 0.05.
b. The problem with repeating the analysis from part (a) with the new dataset of 480 observations is that the data is no longer independent due to repeated measures. A different statistical test that takes into account repeated measures, such as a repeated measures ANOVA, would need to be used to properly analyze this new dataset.
a. To test whether the data suggests that mice show a preference towards one paw over the other, we can use a chi-squared goodness of fit test. The null hypothesis for this test is that there is no preference towards one paw over the other, while the alternative hypothesis is that there is a preference towards one paw over the other.
The observed frequencies are
Right paw: 18 mice
Left paw: 30 mice
We can calculate the expected frequencies assuming no preference using the formula:
Expected frequency = (total number of observations * probability of right paw use) or (total number of observations * probability of left paw use)
Expected frequency of right paw use = 48 * 0.5 = 24
Expected frequency of left paw use = 48 * 0.5 = 24
We can now calculate the chi-squared test statistic using the formula:
chi-squared = sum((observed frequency - expected frequency)^2 / expected frequency)
Substituting the observed and expected frequencies into the formula:
chi-squared = ((18 - 24)^2 / 24) + ((30 - 24)^2 / 24) = 2.5
Using a chi-squared distribution table with one degree of freedom and a significance level of 0.05, the critical value is 3.84.
Since the calculated chi-squared value (2.5) is less than the critical value (3.84), we fail to reject the null hypothesis. Therefore, we do not have enough evidence to conclude that mice show a preference towards one paw over the other.
b. The problem with repeating the analysis from part (a) with the new dataset of 480 observations is that the data is no longer independent. Since each mouse was observed 10 times, the observations for each mouse are not independent, and treating them as such would violate the assumptions of the chi-squared goodness of fit test. To properly analyze this new dataset, a different statistical test that takes into account repeated measures, such as a repeated measures ANOVA, would need to be used.
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The given question is incomplete, the complete question is:
Do mice show a preference for using one paw over the other? Attempting to answer this question, a researcher tested a random sample of 48 mice and observed which forepaw (right or left) the mice used to retrieve food from a narrow tube. Eighteen of the mice used their right paw, while the rest used their left paw.
a. Does this data suggest that mice show a preference towards one paw over the other? Carry out the appropriate test and include all steps for full credit. Given: critical value = 3.84
b. During the experiment, the researcher noticed that individual mice sometimes switched which forepaw they used to retrieve the food. So she repeated this study on the same 48 mice, but measured each one 10 times and she recorded which forepaw was used each time. What would be the problem with repeating the analysis ran in part (a) with her new dataset of 480 observations? Be specific.
Willie runs 5 miles in 25 minutes. If Willie runs at the same rate, how many miles can he run in 30 minutes?
Given Data:
Number of miles covered in 25 minutes = 5
To Find:
Number of miles covered in 30 minutes = ?
Solution:
Number of miles covered per minutes = 5/25 (rate=distance/time)
=0.2 miles/minute
Number of miles covered in 30 minutes = 0.2*30
=6 miles
rent wheels keeps track of the types of vehicles that are rented each week. the table below shows the results from this week. type of vehicle number rented suv 31 truck 8 minivan 14 sedan 27 based on the data, what is the probability that the next vehicle to be rented will be a minivan?
The probability that the next vehicle to be rented will be a minivan from all 80 vehicles for rented wheels is equals to the 7/40.
We have, Rent wheels keeps track of the types of vehicles that are rented each week. The above table shows the results from this week and type of vehicle number.
Number of SUV vehicle = 31
Number of truck vehicle = 8
Number of minivan vehicle = 14
Number of sedan vehicle = 27
So, total number of vehicles or outcomes
= 80
Probability is defined as chances of occurrence of an event. It is calculated by dividing the favourable responses to the total number of possible outcomes or responses. Let E be an event that rented vehicle is minivan. The probability that the next vehicle to be rented will be a minivan, P(E) = 14/80
=> P(E) = 7/40
Hence, the required probability is 7/40.
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Complete question:
The above figure completes the question. rent wheels keeps track of the types of vehicles that are rented each week. the table below shows the results from this week. type of vehicle number rented suv 31 truck 8 minivan 14 sedan 27 based on the data, what is the probability that the next vehicle to be rented will be a minivan?
Eliza gets paid $3,000 per month she pays $750 a month for rent. what percent of her monthly pay goes to rent
Answer: 25%
Step-by-step explanation:
Find the value of x for this problem
After answering the presented question, we can conclude that Pythagorean theorem Therefore, the value of x is 15.
what is Pythagorean theorem?The Pythagorean theorem is a fundamental mathematical principle that describes the relationship between the sides of a right triangle. It states that the sum of the squares of the other two sides' lengths equals the square of the hypotenuse's length (the side opposite the right angle). The mathematical theorem is as follows: c2 = a2 + b2 Where "c" denotes the hypotenuse length and "a" and "b" represent the lengths of the other two sides, known as the legs.
Pythagorean theorem
[tex]x^2 = 12^2 + y^2\\y/x = 3/5\\y = (3/5)x\\x^2 = 12^2 + (3/5)^2 x^2\\(1 - (3/5)^2) x^2 = 12^2\\(1 - 9/25) x^2 = 12^2\\(16/25) x^2 = 12^2\\x^2 = (25/16) * (12^2)\\x = (5/4) * 12 = 15\\[/tex]
Therefore, the value of x is 15.
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C is the midpoint of AB, D is the midpoint of AC, E is the midpoint of AD, F is the midpoint of ED, G is the midpoint of
EF, and H is the midpoint of DB. If DC = 56, what is GH?
When C is the midpoint of AB, D is the midpoint of AC, E is the midpoint of AD, F is the midpoint of ED, G is the midpoint of EF, and H is the midpoint of DB, the length of GH is 77.
How to calculate midpoint for the given problem?
Since D is the midpoint of AC, we know that AD = DC = 56/2 = 28.
Similarly, E is the midpoint of AD, so AE = ED = 28/2 = 14.
Now, F is the midpoint of ED, so EF = ED = 14.
Next, G is the midpoint of EF, so EG = GF = 14/2 = 7.
Finally, H is the midpoint of DB, so DH = HB = AB/2.
Since C is the midpoint of AB, we have AB = 2AC = 2DC = 2*56 = 112.
Therefore, DH = HB = AB/2 = 112/2 = 56.
Now, we can see that GH = GD + DH = (GE + ED) + HB = (7 + 14) + 56 = 77.
Therefore, GH = 77.
The given problem involves a set of line segments and their midpoints. We can use the concept that the midpoint of a line segment divides it into two equal parts. Using this concept, we can determine the lengths of various line segments in terms of the given length DC.
Then we can apply this information to find the length of GH by adding up the appropriate line segments. Specifically, we use the fact that GH is the sum of GD and DH, which in turn are the sum of several mid-segment lengths that can be determined based on the given information. Following this logic, we can calculate that GH is equal to 77.
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A rectangular painting is 42 cm wide and 51 cm long what is the perimeter of this painting
The perimeter of rectangular painting which is 42 cm wide and 51 cm long is 186 cm.
Perimeter of rectangle = 2×(l + b)
Perimeter = 2 × 93
Perimeter = 186 cm
Perimeter is the total length of the boundary of any closed shape. Let’s try and understand this using an example. For example, you have a huge square-shaped farm. Now, in order to save your farm from street animals, you decide to fence it. If you know the length of one side of the farm, you just have to multiply it by 4 in order to find the total length of the boundary of the farm. There are many such instances where we might be using the concept of finding perimeter without even knowing about it.
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Jenna mixed two shades of purple paint. For Purple People Eater, she used 3 pints of red paint for every 5 pints of blue paint. For Purple Majesty, she used a blue to red paint ratio of 7 to 4 . Compare the ratios of red paint to blue paint.
The ratio for Purple People Eater can be expressed in fraction form as
.
The ratio for Purple Majesty can be expressed in fraction form as
.
To compare these fractions, rewrite them with a common denominator of
.
The ratio of red paint to blue paint in Purple People Eater is
the ratio in Purple Majesty.
Answer:
Step-by-step explanation:
The ratio for Purple People Eater can be expressed in fraction form as
✔ 3/5
.
The ratio for Purple Majesty can be expressed in fraction form as
✔ 4/7
.
To compare these fractions, rewrite them with a common denominator of
✔ 35
.
The ratio of red paint to blue paint in Purple People Eater is
✔ greater than
the ratio in Purple Majesty.
An angle measure 64 more than the measure of its supplemental angle. what is the measure of each angle.
Answer:
Original angle = 122*
Supplementary angle = 58*
Step-by-step explanation:
Take the formula and do a little algebra.
x + (x + 64) = 180
Subtract 64 from both sides
x + x = 116
Combine the x's
2x = 116
Divide both side by 2
x = 58
Hope this helps!!
Good luck and mark me brianliest!!
Hii, can someone help me solve this? Thankyou
1)
a) the mean is µ = 2.5.
b) the variance is σ² = 1.25.
c) Standard Deviation σ = 1.12
2)
a) the mean is µ = 6.40.
b) the variance is σ² = 2.12
c) Standard Deviation σ = 1.46.
1) To find the mean (µ), we can multiply each value of x by its corresponding probability (P(x)), and then add up the products: (See attached table I)
Sum | 1 | 2.5
Therefore, the mean is µ = 2.5.
To find the variance (σ²), we need to find the squared differences between each value of x and the mean, multiply each squared difference by its corresponding probability, and then add up the products:
See table II
Sum | 1 | | 5 | 1.25
Therefore, the variance is σ² = 1.25.
Finally, to find the standard deviation (σ), we can take the square root of the variance:
σ = √(σ²) = √(1.25) ≈ 1.12.
2)
To find the mean (µ), we can multiply each value of x by its corresponding probability (P(x)), and then add up the products: See table III...
Sum | 1.00 | 6.40
Therefore, the mean is µ = 6.40/1.00 = 6.40.
To find the variance (σ^2), we need to find the squared differences between each value of x and the mean, multiply each squared difference by its corresponding probability, and then add up the products: See table IV
Sum | 1.00 | | 40.80 | 2.1200
Therefore, the variance is σ² = 2.1200.
Finally, to find the standard deviation (σ), we can take the square root of the variance:
σ = √(σ²) = √(2.1200) ≈ 1.46.
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what is the value of x?
Answer:
x=40
Step-by-step explanation:
alternate angles are equal so the one of the triangles angles r 50 then 50 plus 90, in a right angled triangle, adds to 140
180 minus 140 is 40
helpppp
need the answer asap
The key features of the graph are (a) x-int = 5 and y-int= 5, HA; y = 0.5 and VA: x = 0.5 and hole = (-2, 7/5)
Identifying the key features of the graphThe x-intercept of a graph is the point at which the graph intersects the x-axis while the y-intercept of a graph is the point at which the graph intersects the y-axis.
From the given graph, we have the intercepts to be
x-intercept = 5 and y-intercept = 5
The horizontal asymptote of a graph is a horizontal line that the graph approaches as x approaches positive or negative infinity
While the vertical asymptote is a vertical line that the graph approaches as x approaches a certain value.
From the given graph, we have the asymptote to be
Horizontal; y = 0.5 and Vertical: x = 0.5
A hole in a graph refers to a point on the graph where the function is undefined but can be made continuous by removing the point and filling the gap with a single point.
From the given graph, we have the hole to be
Hole = (-2, 7/5)
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mmon Core Grade 8 Algebra I A
Quiz
Active
Which graph has a rate of change of zero?
Mark this and return
Answer:
A line has slope of zero when it is a horizontal line.
a nurse annual gross salary is $29460 he contributes 2% of his salary to a medical scheme.His contribution to the medical scheme is a non taxable allowance
(a) The total amount of the nurse's annual salary that is not taxed is $6644. (b) His annual taxable salary is $22,226.8. (c) The tax he pays annually is $5856.7.
(a) To find the total amount of the nurse's annual salary that is not taxed, we need to add up all of the non-taxable allowances:
$37 x 12 (months) = $444 for national insurance
$3000 for personal allowance
$1800 for his wife
$1400 for his children
Total non-taxable allowances = $444 + $3000 + $1800 + $1400 = $6644
Therefore, the nurse's total non-taxable salary is $6644.
(b) To find the nurse's annual taxable salary, we need to subtract his non-taxable allowances from his gross salary:
Annual taxable salary = $29,460 - 2% x $29,460 - $6644
Annual taxable salary = $29,460 - $589.2 - $6644
Annual taxable salary = $22,226.8
Therefore, the nurse's annual taxable salary is $22,226.8.
(c) To calculate the nurse's annual tax, we need to apply the income tax rates:
Tax on first $4000 at 20% = $800
Tax on remainder at 25% = 0.25 x ($22,226.8 - $4000) = $5056.7
Total annual tax = $800 + $5056.7 = $5856.7
Therefore, the nurse pays an annual tax of $5856.7.
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Complete question is attached below
What is the answer?
The sequence of transformation that makes the shapes congruent is rotation then transformation
From the question, we have the following parameters that can be used in our computation:
The figures
From the figure, we have the shapes to have different orientation but same size
This means that one of the shapes is rotated to form the other
In this case, the rotation is a 90° clockwise
Followed by the rotation is a translation transformation
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a spinner has sections labeled w , x , y , and z . the faces of a number cube are labeled 1, 2, 3, 4, 5, and 6. what is the total number of possible outcomes of 1 spin of the arrow on the spinner and 1 roll of the number cube?
The total number of possible outcomes of 1 spin of the arrow on the spinner and 1 roll of the number cube is 24.
Here's how to find the solution:
First, determine the number of outcomes for spinning the arrow on the spinner. There are four sections labeled W, X, Y, and Z. Therefore, the number of outcomes for spinning the arrow is 4.Next, determine the number of outcomes for rolling the number cube. There are six faces labeled 1, 2, 3, 4, 5, and 6. Therefore, the number of outcomes for rolling the number cube is 6.To find the total number of possible outcomes of 1 spin of the arrow on the spinner and 1 roll of the number cube, multiply the number of outcomes for spinning the arrow by the number of outcomes for rolling the number cube:4 × 6 = 24.Therefore, the total number of possible outcomes of 1 spin of the arrow on the spinner and 1 roll of the number cube is 24.
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HELP PLEASE!!
The monthly rent for the first house is $1,190,
and the Bainters can expect it to increase 1.7%
every year. The Bainters want to calculate their monthly rent over time.
How much will the monthly rent be after 1
year?
Round your answer to the nearest whole dollar if necessary.
Remember to write down and keep your calculations.
We must raise the $1,190 starting rent by 1.7% in order to determine the monthly rent after a year.
We can begin by figuring out how much the rise will be:
Amount of increase: 1.7% of $1,190 multiplied by 0.017 times equals $20.23. (rounded to the nearest cent)
We must multiply the initial rent by this sum in order to determine the new monthly rate:
New rate is $1,190 plus $20.23 per month, or $1,210.23. (rounded to the nearest cent)
As a result, the rate will be $1,210 per month after a year.
What is the starting point?When the input parameter value is 0, the output of a function is its starting value, or 0. The first number would be the amount of money you had to start with on day 0 if your function, for instance, was tracking the sum of money you earned over a specific period of time.
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The monthly rent after 1 year will be $1,210.
What is amount ?"Amount" is a general term that refers to the quantity, value, or total of something. It can be used in different contexts to refer to different things.
To calculate the monthly rent after 1 year, we need to find the 1.7% increase and add it to the initial rent.
First, we need to calculate the amount of the increase:
Increase = 1.7% of $1,190
Increase = 0.017 × $1,190
Increase = $20.23 (rounded to two decimal places)
Next, we can find the new monthly rent by adding the increase to the initial rent:
New rent = $1,190 + $20.23
New rent = $1,210.23 (rounded to two decimal places)
Therefore, the monthly rent after 1 year will be $1,210.
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What is the area of the regular polygon?
(Round to one decimal place as needed)
PLS PLS PLS HELP THIS IS DUE TODAY (geometry btw)
The area of the polygon is 6. 9m²
How to determine the area of the polygonFrom the information given, we have that the regular polygon takes the shape of a triangle.
The formula for calculating the area of a triangle is expressed with the equation;
A = 1/2bh
Such that the parameters in the formula are;
A is the area of the triangle.b is the base of the triangle.h is the height of the triangle.Area of a equilateral triangle is;
= √3/4a²
Then,
= √3/4 (4)²
= √3 × 4
= 6. 9m²
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The graph of y=x^2 is shown in the figure. What is the average rate if change of this function over the interval -1
Answer:
The average rate of change of the function h(x) over the interval between 2 points (a , h(a)) and (b ,h(b)) is the
slope of the secant line
connecting the points.
It is found using.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
∣
∣
∣
2
2
h
(
b
)
−
h
(
a
)
b
−
a
2
2
∣
∣
∣
−−−−−−−−−−−−−−−−
h
(
b
)
=
h
(
2
)
=
2
2
+
3
(
2
)
−
1
=
9
h
(
a
)
=
h
(
0
)
=
−
1
⇒
9
−
(
−
1
)
2
−
0
=
10
2
=
5
This means that the average of all the slopes of tangents between (2 ,9) and (0 ,-1) is 5
Are these triangles identical? Explain your reasoning
No, the triangles are not identical
What are identical triangles?Identical triangles are two triangles that have the exact same
shape and size.In other words, all corresponding sides and angles are congruent (equal).
Identical triangles can be thought of as the same triangle, just in different positions or orientations in space.
The triangles are similar triangles but not identical triangles as the sides are not equal
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This graph shows the relationship between numbers of cookies (c) sold and profit earned (p).
Enter an equation to represent the number of cookies sold and profit earned.
Based οn the infοrmatiοn prοvided, we can see that the prοfit earned (p) is prοpοrtiοnal tο the number οf cοοkies sοld (c).
We can write the equatiοn as:
p = 0.25c
This equatiοn means that fοr every cοοkie sοld, the prοfit earned is $0.25. We can alsο write this equatiοn in slοpe-intercept fοrm as:
y = mx + b
where y represents prοfit earned (p), x represents the number οf cοοkies sοld (c), m represents the slοpe (0.25 in this case), and b represents the y-intercept (the value οf y when x is 0, which is 0 in this case).
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What exponential function can be used to determine the number of transistors in a car that doubles every TWO years? The number starts at 4100. For this function to work, we should be able to find the amount of transistors in a car after 5 years (or any odd number of years)
Answer:
[tex]f(t) = 4100( {2}^{ \frac{t}{2} }) [/tex]
JANO
5. Which of the following best describes how to calculate the periodic interest rate for most
credit cards?
a.
Divide the annual interest rate by 12
Divide the annual interest rate by 52
C.
Divide the annual interest rate,by 365
d. Divide the number of days in the billing cycle by the annual interest rate
b.
The best way to calculate the periodic interest rate for most credit cards is (a) to divide the annual interest rate by 12.
What is interest?Interest is a fee charged by a lender to a borrower for the use of borrowed money or assets. It is typically expressed as a percentage of the principal amount borrowed or invested, and it is calculated over a specific period of time. Interest can be charged on loans, credit cards, mortgages, bonds, and other financial instruments. Interest can be positive or negative, depending on whether the borrower or lender is paying or receiving the interest. Positive interest is also known as a return or yield, and it is earned by investors who lend money or invest in financial instruments. Negative interest, on the other hand, is a fee charged by lenders to borrowers, and it is typically associated with negative interest rates or deflationary economic conditions.
Here,
Most credit cards compound interest on a monthly basis, which means that the interest is calculated and added to the balance each month. Therefore, to calculate the monthly or periodic interest rate, we need to divide the annual interest rate by 12.
For example, if the annual interest rate on a credit card is 18%, then the monthly or periodic interest rate would be:
Periodic interest rate = 18% / 12
= 1.5%
Therefore, the correct answer is (a) Divide the annual interest rate by 12.
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Ariana 0.45 of the groups and Alex ate 35% of them if there was 18 grapes left how many were in the container at first
The original number of grapes present in the container was 50.35.
To determine the number of grapes Ariana ate.
then, we'll find the number of grapes Alex ate.
Finally, we'll use the remaining grapes to determine the original number of grapes present in the container.
Let us solve the problem now.
Let the original number of grapes be x. Ariana ate 0.45 of them.
Number of grapes Ariana ate = 0.45x
Alex ate 35% of the remaining grapes after Ariana ate.
Number of grapes Alex ate = 35% of (x - 0.45x)= 35/100(x - 0.45x)= 0.35(0.55x)= 0.1925x
The number of grapes remaining = 18.Number of grapes remaining after both Ariana and
Alex ate = x - 0.45x - 0.1925x= 0.3575x= 18
Hence, we can determine that:0.3575x = 18x = 18/0.3575x = 50.35
The original number of grapes present in the container was 50.35.
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What is the surface area of a cube that measures 5 inches on each side? 75 in 2 50 in 2 150 in 2 25 in 2
Answer:
The surface area of a cube can be calculated using the formula 6s^2, where s is the length of one of the sides of the cube.
In this case, s = 5 inches, so the surface area would be:
6(5 in)^2 = 6(25 in^2) = 150 in^2
Therefore, the surface area of the cube that measures 5 inches on each side is 150 in^2. So the answer is c) 150 in^2.
Step-by-step explanation:
To find the surface area of a cube, we use the formula:
Surface area = 6s^2
where s is the length of one of the sides of the cube.
In this case, the length of each side of the cube is given as 5 inches.
Substituting this value into the formula, we get:
Surface area = 6 x 5^2
Surface area = 6 x 25
Surface area = 150 square inches
Therefore, the surface area of the cube is 150 square inches.
(1 point) an open rectangular box has volume 32 cm3 . what are the lengths of the edges giving the minimum surface area?
So, the length of open rectangle box is 4cm, width is 4cm, and height is 2cm, with Surface Area of 48cm^2.
it is an Open box, so there is no top
SA (Surface area) = 2*l*h + 2*w*h + l*w
Volume(V) = l*w*h = 32cm^3
h = 32/(w*l)
SA = 2*l*32/(w*l) + 2*w*32/(w*l) + l*w
SA = 64/w + 64/l + l*w
Find minimum SA: take partial derivatives to get critical point(s)
SAw = -64/w^2 + l
SAl = -64/l^2 + w
Both the partials have to be 0, so...
0 = -64/w^2 + l and 0 = -64/l^2 + w
64/w^2 = l
0 = -64/(64/w^2)^2 + w (plug into second equation)
0 = -w^4/64 + w
0 = w(1-w^3/64)
1 = w^3/64 or 0 = w (impossible answer)
64 = w^3
4 = w
Plug w back into 64/w^2 = l
64/4^2 = l
4 = l
Plug w and l back into h = 32/(w*l)
h = 32/(4*4)
h = 2
The Surface Area:2*l*h + 2*w*h + l*w
SA = 2*4*2 + 2*4*2 + 4*4
SA = 48
So the answer is length = 4cm, width = 4cm, and height = 2cm, with Surface Area of 48cm^2
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A flashlight is projecting a triangle onto a wall, as shown below.
The original triangle and its projection are similar. What is the missing length n on the projection?
After answering the given question, we can conclude that As a result, the triangle projection's lacking length is 8.
What precisely is a triangle?A triangle is a closed, double-symmetrical structure composed of three line segments known as sides that meet at three locations known as vertices. Triangles are distinguished by their edges and angles. Triangles can be equilateral (all groups equal), isosceles, or scalene based on their edges. Triangles are classified as acute (all angles are less than 90 degrees), okay (one angle is identical to 90 degrees), or orbicular (all angles are higher than 90 degrees) (all angles greater than 90 degrees). The area of a triangle can be calculated using the formula A = (1/2)bh, where an is the neighbourhood, b is the triangle's base, and h is the triangle's height.
Because the initial triangular and its projection are comparable, the sides that correspond to them are proportional. Let x be the length of the initial triangle's side equivalent to n. Then, given that the extension of the length x side has length 8, we can write:
x/n = (initial triangle side length)/(projected triangle side length) = x/8
When we solve for n, we get:
n = (8x)/x = 8
As a result, the projection's lacking length is 8.
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For maximum efficiency if factory must have at least 100 workers but no more than 200 workers on a shift in the factory also must manufacture at least 30 units per worker. A) let X be the number of workers and let YB the number of units rate for an inequality is expressing the conditions in the problem given. B) graph the systems of inequalities. (Not required) C) List three possible solutions.
Therefore , the solution of the given problem of inequality comes out to be 150 units per employee, to satisfy the production demand.
What is an inequality ?Despite the fact that algebra lacks a comparable symbol, its distinction can be expressed by a pair or collection for numbers. Equilibrium is typically followed by equity. The ongoing disparity in norms is what causes inequality. Disparity and fairness are not synonymous. Even though the components are typically not connected or situated closely together, that was our most popular symbol.
Here,
A) Assume Y is the number of units created and X is the number of employees. The following inequalities represent the circumstances in the issue:
=> 100 ≤ X ≤ 200 (the factory must have at least 100 but no more than 200 workers)
=> Y ≥ 30X (the factory must manufacture at least 30 units per worker)
B) The lines 100, 200, and 30X would form the borders of the darkened area.
C) The following three options meet the set of inequalities:
=> X = 100, Y = 3000
=> X = 150, Y = 4500
=> X = 200, Y = 6000
The factory in the second solution employs 150 people and generates 4500 units, or 150 units per employee, to satisfy the production demand.
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