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Vadim26 [7]
3 years ago
7

Find the dimensions of a rectangle with perimeter 112 m whose area is as large as possible. step 1 if a rectangle has dimensions

x and y, then we must maximize the area a = xy. since the perimeter is 2x + 2y = 112, then y = 56
Mathematics
1 answer:
Veronika [31]3 years ago
7 0


Please use the solution below:
Let P = perimeter, A = area
As provided above, a = xy and 
We know that the formula to solve for the perimeter of a rectangle is P = 2x + 2y. Using the given 112m, we can solve the perimeter using the formula:
112 = 2x + 2y
56 = x + y

x = 56-y or y = 56-x
Let's solve the perimeter in terms of y using the formula below:
A = (56-y)(y)
Find the derivative of A = 56-y^2 to get the value of y.
dA/dy = 56-2y = 0
y = 56/2
y = 28
To find X, substitute the value of y in the equation x = 56 - y. 
x = 56 - 28
Therefore, x = 28.
We can conclude that the figure is not a rectangle but a square.
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A sample of 125 pieces of yarn had mean breaking strength 6.1 N and standard deviation 0.7 N. A new batch of yarn was made, usin
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Answer:

The 90% confidence interval for the difference in mean breaking strength between the two types of yarn is (0.08N, 0.52N).

Step-by-step explanation:

Before building the confidence interval, we need to understand the central limit theorem and subtraction of normal variables.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean \mu and standard deviation \sigma, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

Subtraction of normal variables:

When we subtract two normal variables, the mean is the difference of the means, while the standard deviation is the square root of the sum of the variances.

A sample of 125 pieces of yarn had mean breaking strength 6.1 N and standard deviation 0.7 N.

This means that \mu_1 = 6.1, s_1 = \frac{0.7}{\sqrt{125}} = 0.0626

In a sample of 75 pieces of yarn from the new batch, the mean breaking strength was 5.8 N and the standard deviation was 1.0 N.

This means that \mu_2 = 5.8, s_2 = \frac{1}{\sqrt{75}} = 0.1155

Distribution of the difference:

\mu = \mu_1 - \mu_2 = 6.1 - 5.8 = 0.3

s = \sqrt{s_1^2+s_2^2} = \sqrt{0.0626^2+0.1155^2} = 0.1314

Confidence interval:

We have that to find our \alpha level, that is the subtraction of 1 by the confidence interval divided by 2. So:

\alpha = \frac{1 - 0.9}{2} = 0.05

Now, we have to find z in the Ztable as such z has a pvalue of 1 - \alpha.

That is z with a pvalue of 1 - 0.05 = 0.95, so Z = 1.645.

Now, find the margin of error M as such

M = zs

M = 1.645*0.1314 = 0.22

The lower end of the interval is the sample mean subtracted by M. So it is 0.3 - 0.22 = 0.08N

The upper end of the interval is the sample mean added to M. So it is 0.3 + 0.22 = 0.52N

The 90% confidence interval for the difference in mean breaking strength between the two types of yarn is (0.08N, 0.52N).

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Using Triangle Congruence Theorems
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According to the SAS rule, two triangles are said to be congruent if any two sides and any angle between the sides of one triangle are equal to the corresponding two sides and angle between the sides of the second triangle.

SSS - Side Side Side Rule

According to the SSS rule, two triangles are said to be congruent if all three sides of one triangle are proportional to the size three sides of the second triangle.

AAS - Angle Angle Side Rule
Angle-Angle-Side is abbreviated as AAS. The triangles are said to be congruent when two angles and a non-included side of one triangle match the comparable angles and sides of another triangle.

ASA - Angle Side Angle rule

According to the ASA rule, two triangles are said to be congruent if any two angles and the side contained between the angles of one triangle are proportional to the size two angles and side included in between angles of the second triangle.

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