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pashok25 [27]
3 years ago
11

The length of a rectangular picture frame is twice the width. The frame has a perimeter of 96 inches. Write and

Mathematics
2 answers:
Simora [160]3 years ago
4 0

Answer:

Length = 16 inches

Width = 32 inches

Step-by-step explanation:

Width of the frame is represented by x and length by y inches. It is given that length of the frame is twice the width. This means y is twice of x. So, we can write the equation as:

y = 2x                                      Equation 1

Perimeter of a rectangle is defined as:

Perimeter =  2 ( Length + Width )

Since, perimeter of the picture frame is equal to 96, we can write the equation as:

2(x + y) = 96                          Equation 2

Substituting the value of y from Equation 1 in Equation 2, we get:

2(x + 2x) = 96

2(3x) = 96

6x = 96

x = 16

Substituting the value of x in Equation 1, we get:

y = 2x = 2(16) = 32

This means, the length of the rectangular picture frame is 16 inches and its width is 32 inches.

Vladimir [108]3 years ago
3 0

Answer:

Width is 16 inches and length is 32 inches.

Step-by-step explanation:

Given: The length of a rectangular picture frame is twice the width. The frame has a perimeter of 96 inches.

To find: Length and width

Solution:

Let the x represent the width of the frame and y represent the length of the frame.

We have,

The perimeter of the frame is 96 inches.

length of the frame is twice its width, so y=2x

Now,

\text{Perimeter}=2(l+w)

\text{Perimeter}=2(y+x)

\text{Perimeter}=2(2x+x)

\text{Perimeter}=6x

\implies 6x=96

\implies x=16

y=2(16)=32

So, width of the frame is 16 inches, and length of the rectangle is 32 inches

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A certain type of aluminum screen has, an average, one flaw in a 100-foot roll. Assume the flaw distribution approximately follo
julsineya [31]

Answer:

a) 0.9197 = 91.97% probability that a 100-foor roll has at most 2 flaws.

b) 0.1074 = 10.74% probability that there are exactly 3 rolls that have no flaws in them.

c) 0.0394 = 3.94% probability that the 15th roll he inspected was the 3rd one that have no flaws in it.

Step-by-step explanation:

To solve this question, we need to understand the Poisson and the Binomial distribution.

Poisson distribution:

In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:

P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}

In which

x is the number of sucesses

e = 2.71828 is the Euler number

\mu is the mean in the given interval.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}

In which C_{n,x} is the number of different combinations of x objects from a set of n elements, given by the following formula.

C_{n,x} = \frac{n!}{x!(n-x)!}

And p is the probability of X happening.

(a) Find the probability that a 100-foor roll has at most 2 flaws.

A certain type of aluminum screen has, an average, one flaw in a 100-foot roll, which means that \mu = 1. Only one roll means that the Poisson distribution is used.

This is:

P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)

In which

P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}

P(X = 0) = \frac{e^{-1}*1^{0}}{(0)!} = 0.3679

P(X = 1) = \frac{e^{-1}*1^{1}}{(1)!} = 0.3679

P(X = 2) = \frac{e^{-1}*1^{2}}{(2)!} = 0.1839

P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) = 0.3679 + 0.3679 + 0.1839 = 0.9197

0.9197 = 91.97% probability that a 100-foor roll has at most 2 flaws.

(b) Suppose that I bought 10 200-foot rolls, find the probability that there are exactly 3 rolls that have no flaws in them.

Two parts.

Probability that a single 200-foot roll has no flaw.

This is P(X = 0), Poisson(single 200-foot roll) when \mu = \frac{200*1}{100} = 2. So

P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}

P(X = 0) = \frac{e^{-2}*2^{0}}{(0)!} = 0.1353

0.1353 probability that a single 200-foot roll has no flaw.

Probability that on 10 200-foot rolls, 3 have no flaws.

Multiple 200-foot rolls means that we use the binomial distribution.

0.1353 probability that a single 200-foot roll has no flaw means that p = 0.1353

10 200-foot rolls means that n = 10

We want P(X = 3). So

P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}

P(X = 3) = C_{10,3}.(0.1353)^{3}.(0.8647)^{7} = 0.1074

0.1074 = 10.74% probability that there are exactly 3 rolls that have no flaws in them.

(c) What is the probability that the 15th roll he inspected was the 3rd one that have no flaws in it.

0.1353 probability that a single 200-foot roll has no flaw means that p = 0.1353

2 with flaws in the first 14, which is P(X = 2) when n = 14

The 15th has no flaw, with probability of 0.1353. So

P = 0.1353*P(X = 2) = 0.1353*(C_{14,2}.(0.1353)^{2}.(0.8647)^{12}) = 0.1353*0.2911 = 0.0394

0.0394 = 3.94% probability that the 15th roll he inspected was the 3rd one that have no flaws in it

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Answer:

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Step-by-step explanation:

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We have, r = x, R = (1/2)x, h= ? (height of the cone)

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Answer:

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