Find the sum: (2x2 + x + 3) + (3x2 + 2x + 1).

A cardboard box without a lid is to have a volume of 8,788 cm3. Find the dimensions that minimize the amount of cardboard used.

Montano1993 [528]

Answer:

x = y = 26 cm; z = 13 cm

Step-by-step explanation:

The generic solution for the minimum material in an open-top box is that the box is square and half as high as it is wide. It is half a cube of twice the volume.

The dimensions of the square base are ...

∛(2·8788 cm³) = 26 cm

Then the height is half that, or 13 cm.

x = y = 26 cm; z = 13 cm

_____

If you need to see the development, you can use the method of Lagrange multipliers to find the minimum area for the given volume;

area = xy +2(xz +yz)

volume = xyz = 8788

We require each of the partial derivatives of L with respect to x, y, z, and λ to be zero.

L = xy +2(xz +yz) +λ(xyz -8788)

partial with respect to x: 0 = y+2z +λyz

partial with respect to y: 0 = x +2z +λxz

partial with respect to z: 0 = 2x+2y +λxy

partial with respect to λ: 0 = xyz -8788

From the first two equations, we have ...

λ = (y +2z)/(yz) = 1/z +2/y

λ = (x +2z)/(xz) = 1/z +2/x

Equating these expressions for λ, we find ...

1/z +2/y = 1/z +2/x ⇒ x = y

The third equation then tells us ...

λ = (2x +2y)/(xy) = 2/y +2/x

Comparing this to either of the first two expressions for λ, we see ...

1/z +2y = 2/x +2/y ⇒ z = x/2

This is the result we started the answer with:

x = y = 2z = ∛(2·8788 cm³)

7 0

3 years ago

Simplify ((1/x)-(1/y))/((1/x)+(1/y))

In-s [12.5K]

<span>Simplify ((1/x)-(1/y))/((1/x)+(1/y))</span>

y-x
______

y+x

7 0

3 years ago

Assume the average height for American women is 64 inches with a standard deviation of 2 inches. A sorority on campus wonders if

satela [25.4K]

Answer:

a) s = 0.4

b) Z = 2.5

c) 99th percentile.

d) The larger sample size would lead to a smaller margin of error, which would lead to a higher z-score and a increased percentile.

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$ , the z-score of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem establishes 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 $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$ .

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

Assume the average height for American women is 64 inches with a standard deviation of 2 inches

This means that $\mu = 64, \sigma = 2$

A) Calculate the standard error for the distribution of means.

Sample of 25 means that $n = 25$ , so $s = \frac{2}{\sqrt{25}} = 0.4$ .

B) Calculate the z statistic for the sorority group.

Sample mean of 65 means that $X = 65$ .

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem

$Z = \frac{X - \mu}{0.4}$

$Z = \frac{65 - 64}{0.4}$

$Z = 2.5$

C) What is the approximate percentile for this sample? Enter as a whole number.

Z = 2.5 has a p-value of 0.9938, so 0.99*100 = 99th percentile.

D) If the sorority actually had 36 members (still with an average of 65 inches), would you expect the percentile value to increase or decrease? why?

The larger sample size would lead to a smaller margin of error, which would lead to a higher z-score and a increased percentile.

3 0

2 years ago

Study the figure.

tia_tia [17]

Answer:

So we know the formula to calculate the area of the circular sector:

S=(r^2*π*a)/306°=

(5^2*3.14*40°)/360°= (1000*3.14) /360=8.72cm^2 so the right alternative should be the first one ,A.

Step-by-step explanation:

r - radius of the circle

a - corner of the circular sector

S - surface

4 0

3 years ago

How do you round 9.246 to the nearest tenth?

Flauer [41]

You look at the hundreths place (4) and if it is 5 or more you round the tenths place up. so the number is rounded to 9.2

3 0

3 years ago