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Question 131:
To get from the pooi to home, Fred can either take a bike path through the rectangular park or ride his bike along the two sides of the park.
What measurement is closest to the distance of the bike path?
A.
2.45mi
C.
1.45mi
B.
3.4mi
D.
0.7mi

Background $\newline$ Layout $\newline$ Theme $\newline$ Trans $\newline$ Question $131$ : $\newline$ To get from the pooi to home, Fred can either take a bike path through the rectangular park or ride his bike along the two sides of the park. $\newline$ What measurement is closest to the distance of the bike path? $\newline$ A. $2.45\,\text{mi}$ $\newline$ B. $3.4\,\text{mi}$ $\newline$ C. $1.45\,\text{mi}$ $\newline$ D. $0.7\,\text{mi}$

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Pythagorean Theorem and its converse

Full solution

Q. Background

\newline

Layout

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Theme

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Trans

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Question

131

:

\newline

To get from the pooi to home, Fred can either take a bike path through the rectangular park or ride his bike along the two sides of the park.

\newline

What measurement is closest to the distance of the bike path?

\newline

A.

2.45\,\text{mi}

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B.

3.4\,\text{mi}

\newline

C.

1.45\,\text{mi}

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D.

0.7\,\text{mi}

Identify Layout and Paths: Identify the layout of the park and the paths Fred can take. The park is rectangular, and Fred can either take a direct bike path through the park or ride around two sides of the rectangle.
Calculate Diagonal Length: Assume the dimensions of the park are $x$ miles and $y$ miles for the two sides. The direct path is the diagonal of the rectangle. Use the Pythagorean theorem to find the diagonal: $\text{diagonal}^2 = x^2 + y^2$ .
Estimate Diagonal Length: Without specific values for $x$ and $y$ , we can't calculate the exact diagonal. However, we can estimate based on the answer choices. Assume $x$ and $y$ are such that the diagonal is close to one of the options given: $2.45$ mi, $1.45$ mi, $3.4$ mi, or $0.7$ mi.
Compare Diagonal to Perimeter: If Fred rides around the park, he travels $x + y$ miles. This distance must be reasonable compared to the diagonal. The diagonal should be less than $x + y$ because it's a direct path.
Final Comparison: Compare the diagonal options to the perimeter options. If $x = 2$ mi and $y = 1$ mi, the perimeter is $3$ mi. The diagonal would be $\sqrt{2^2 + 1^2} = \sqrt{5} \approx 2.24$ mi. This is close to one of the answer choices.

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\newline

Can we use the mean value theorem to say the equation

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\newline

1

\newline

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\newline

(B) No, since the average rate of change of

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\newline

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