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3 years ago

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The length of a rectangle is 8 feet more than the width. If the width is increased by 4 feet, and the length is decreased by 5 f

eet, the area remains the same. Find the dimensions of the original rectangle.

1 answer:

SVEN [57.7K]3 years ago

7 0

Answer:

Width = 12 feet

Length = 20 feet

Step-by-step explanation:

The area of a rectangle is A = l*w. Here the length is 8 feet more than the width or 8 + w and the width is w. So the area is A = w(8+w) = 8w + w²

If the dimensions are changed then the expressions will change. The width is increased by 4 feet. This means the width w becomes w + 4. The length is decreased by 5 feet. This means the length becomes 8 + w - 5 = 3 + w. So the area of the new triangle is A = (w+4)(3+w) = w² + 7w + 12.

Since the areas are the same, set them equal to each other and solve for w.

w² + 7w + 12 = 8w + w² Subtract w² from both sides.

7w + 12 = 8w Subtract 7w from both sides.

12 = w

The original width w is 12 feet. This means the original length which was 8 + w = 8 + 12 = 20 feet.

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Simplify by combining like terms: -4x - 2y - 7x - 8y

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Step-by-step explanation:

Solving like terms

-4x - 7x - 2y -8y

-11x - 10y

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Six more than a quotient of a number and three is greater than 14

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Remark

You sure do have to read that a couple of times.

The quotient is an answer after a division takes place So you are dividing x/3

When you have figured out that phrase you add six onto it.

(x/3) + 6 > 14 Subtract 6 from both sides.

(x/3) +6 - 6 > 14 - 6

(x/3) > 8 Now multiply by 3

x > 8*3

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3 years ago

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If 3 quarts of paint are needed for 50 ft of fence, how many quarts are needed for 300 ft of fence?

Zina [86]

Answer:

18

Step-by-step explanation:

First, you would do a proportion

$\frac{3}{50}$ = $\frac{x}{300}$

Then you would cross multiply, getting 50x=900.

After that, you would do 900/50, which is 18.

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3 years ago

Time spent using e-mail per session is normally distributed, with mu equals 11 minutes and sigma equals 3 minutes. Assume that

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Answer:

a) 0.259

b) 0.297

c) 0.497

Step-by-step explanation:

To solve this problem, it is important to know the normal probability distribution and the central limit theorem.

Normal probability distribution

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$ , the zscore 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 pvalue, we get the probability that the value of the measure is greater than X.

Central limit theorem

The Central Limit Theorem estabilishes that, for a random variable X, with mean $\mu$ and standard deviation $\sigma$ , a large sample size can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$

In this problem, we have that:

$\mu = 11, \sigma = 3$

a. If you select a random sample of 25 sessions, what is the probability that the sample mean is between 10.8 and 11.2 minutes?

Here we have that $n = 25, s = \frac{3}{\sqrt{25}} = 0.6$

This probability is the pvalue of Z when X = 11.2 subtracted by the pvalue of Z when X = 10.8.

X = 11.2

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

By the Central Limit Theorem

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

$Z = \frac{11.2 - 11}{0.6}$

$Z = 0.33$

$Z = 0.33$ has a pvalue of 0.6293.

X = 10.8

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

$Z = \frac{10.8 - 11}{0.6}$

$Z = -0.33$

$Z = -0.33$ has a pvalue of 0.3707.

0.6293 - 0.3707 = 0.2586

0.259 probability, rounded to three decimal places.

b. If you select a random sample of 25 sessions, what is the probability that the sample mean is between 10.5 and 11 minutes?

Subtraction of the pvalue of Z when X = 11 subtracted by the pvalue of Z when X = 10.5. So

X = 11

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

$Z = \frac{11 - 11}{0.6}$

$Z = 0$

$Z = 0$ has a pvalue of 0.5.

X = 10.5

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

$Z = \frac{10.5 - 11}{0.6}$

$Z = -0.83$

$Z = -0.83$ has a pvalue of 0.2033.

0.5 - 0.2033 = 0.2967

0.297, rounded to three decimal places.

c. If you select a random sample of 100 sessions, what is the probability that the sample mean is between 10.8 and 11.2 minutes?

Here we have that $n = 100, s = \frac{3}{\sqrt{100}} = 0.3$

This probability is the pvalue of Z when X = 11.2 subtracted by the pvalue of Z when X = 10.8.

X = 11.2

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

By the Central Limit Theorem

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

$Z = \frac{11.2 - 11}{0.3}$

$Z = 0.67$

$Z = 0.67$ has a pvalue of 0.7486.

X = 10.8

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

$Z = \frac{10.8 - 11}{0.3}$

$Z = -0.67$

$Z = -0.67$ has a pvalue of 0.2514.

0.7486 - 0.2514 = 0.4972

0.497, rounded to three decimal places.

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3 years ago

Marcus states that the polynomial expression 3x3 – 4x2y + y2 + 2 is in standard form. Ariel states that it should be y2 – 4x2y +

lora16 [44]

Answer:

Here Ariel is wrong because 3x^3 is higher power than y^2

3x^3 goes first

then the mixed term -4x^2y

then the y^2

then the 2

So, Marcus is right

Standard form means that you write the terms by descending degree. here's main rules how to write polynomial expression in the standard form:

1. write the term with the highest exponent first (terms of third degree in our case 3x^3 );

2. write the terms with lower exponents in descending order (term -4x^2y followed by y^2 );

3. remember that a variable with no exponent has an understood exponent of 1 (here are no such terms);

4. a constant term (a number with no variable) always goes last (term 2).

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