Question

the figure represents a ramp in the shape of a right triangular prism. the dotted line in the ramp shown is the shortest, but steepest, distance, in feet (ft), up the ramp. the dashed line is the longest distance in a straight line up the ramp, but has the least steep slope. how much longer is the distance, in feet, with the least slope than the distance with the greatest slope? (round your answer to the nearest tenth of a foot.)

Explanation:

Step1: Find the length of the shortest (steepest) path

The shortest (steepest) path is the height of the right - triangle cross - section of the prism. Using the Pythagorean theorem for the right - triangle with legs of length 2 ft and 8 ft. Let the height of the right - triangle be $h_1$. Then $h_1=\sqrt{2^{2}+8^{2}}=\sqrt{4 + 64}=\sqrt{68}\approx 8.246$ ft.

Step2: Find the length of the longest (least steep) path

The longest (least steep) path is the length of the hypotenuse of the right - triangular prism along its length. The length of the prism is 24 ft, and the base of the right - triangle cross - section has a hypotenuse of $\sqrt{2^{2}+8^{2}}=\sqrt{68}$ ft. The length of the longest path $h_2$ is the length of the diagonal of the rectangular face with sides 24 ft and $\sqrt{68}$ ft. But we can also consider the path along the length of the prism. The longest path is 24 ft.

Step3: Calculate the difference

Find the difference between the two lengths. Let $d$ be the difference. Then $d = 24-\sqrt{68}\approx24 - 8.246=15.8$ ft.

Answer:

15.8