Question

problems 8-9
a standard city block in manhattan is a rectangle measuring 80 meters by 270 meters.
sol is thinking about cutting diagonally through the park to get from point a to point b.
determine the distance sol would walk by cutting through the park.
if sol walks an average of 1.42 meters per second, how much time will cutting through the park save? round your answer to the nearest second.

Explanation:

First Sub - Question: Determine the distance Sol would walk by cutting through the park.

Step1: Identify the problem type

This is a right - triangle problem where the two legs of the right triangle are the length and width of the rectangular city block, and the diagonal (the path through the park) is the hypotenuse. We can use the Pythagorean theorem, which states that for a right triangle with legs \(a\) and \(b\) and hypotenuse \(c\), \(c=\sqrt{a^{2}+b^{2}}\). Here, \(a = 270\) meters and \(b = 80\) meters.

Step2: Apply the Pythagorean theorem

Substitute \(a = 270\) and \(b = 80\) into the formula \(c=\sqrt{a^{2}+b^{2}}\). First, calculate \(a^{2}\) and \(b^{2}\):
\(a^{2}=270^{2}=270\times270 = 72900\)
\(b^{2}=80^{2}=80\times80 = 6400\)
Then, find the sum of \(a^{2}\) and \(b^{2}\): \(a^{2}+b^{2}=72900 + 6400=79300\)
Finally, find the square root of the sum: \(c=\sqrt{79300}\approx281.6\) (rounded to one decimal place)

Step1: Calculate the distance of the non - diagonal path

The non - diagonal path is the sum of the length and the width of the rectangle. So, the distance \(d_{1}=270 + 80=350\) meters.

Step2: Calculate the time taken for each path

We know that time \(t=\frac{d}{v}\), where \(d\) is the distance and \(v\) is the speed.
For the non - diagonal path: \(t_{1}=\frac{350}{1.42}\approx246.48\) seconds (rounded to two decimal places)
For the diagonal path: \(t_{2}=\frac{281.6}{1.42}\approx198.31\) seconds (rounded to two decimal places)

Step3: Calculate the time saved

The time saved \(\Delta t=t_{1}-t_{2}\). Substitute the values of \(t_{1}\) and \(t_{2}\): \(\Delta t = 246.48-198.31 = 48.17\approx48\) seconds (rounded to the nearest second)

Answer:

The distance Sol would walk by cutting through the park is approximately \(281.6\) meters.

Second Sub - Question: If Sol walks an average of 1.42 meters per second, how much time will cutting through the park save?