Question

1. deriving the pythagorean theorem to find the length of the cable car route

discuss & write: use the points (-6404, 6570) and (90, 9600) to find the vertical and horizontal distances between the two towns. show your work.
discuss & write: use the pythagorean theorem to find the length of the cable - car route. show your work. round your answer to the nearest foot.
discuss & write: the cable car travels at a speed of 704 feet per minute. how many minutes does it take the cable car to get from one town to the other? round your answer to the nearest minute and use the speed equation provided (speed = distance/time).
discuss & write: will the cable car make the one - way trip from auriga to capella in 30 minutes or less? why or why not?

Explanation:

Step1: Calculate horizontal distance

Let $(x_1,y_1)=(- 6404,8570)$ and $(x_2,y_2)=(90,9800)$. The horizontal distance $d_x$ between two points is given by $d_x=\vert x_2 - x_1\vert=\vert90-(-6404)\vert=\vert90 + 6404\vert=6494$.

Step2: Calculate vertical distance

The vertical distance $d_y$ between two points is given by $d_y=\vert y_2 - y_1\vert=\vert9800 - 8570\vert=1230$.

Step3: Use Pythagorean theorem to find length of cable - car route

By the Pythagorean theorem $d=\sqrt{d_x^{2}+d_y^{2}}$. Substitute $d_x = 6494$ and $d_y=1230$ into the formula: $d=\sqrt{6494^{2}+1230^{2}}=\sqrt{42162036+1512900}=\sqrt{43674936}\approx6609$ feet.

Step4: Calculate time taken

We know that speed $s = 704$ feet per minute and distance $d\approx6609$ feet. Using the formula $s=\frac{d}{t}$, we can solve for time $t$. Rearranging gives $t=\frac{d}{s}=\frac{6609}{704}\approx9$ minutes.

Step5: Check if it can make the trip in 30 minutes or less

Since $t\approx9$ minutes and $9<30$, the cable - car will make the one - way trip from Auriga to Capella in 30 minutes or less.

Answer:

Horizontal distance: 6494 feet
Vertical distance: 1230 feet
Length of cable - car route: 6609 feet
Time taken: 9 minutes
Yes, the cable - car will make the one - way trip in 30 minutes or less because it takes approximately 9 minutes.