Question

the ratio of horizontal distance to height of the ramp is 13:1. a builder has a roll of non - slip rubber mat that is 13 feet long. does he have enough rubber to cover the ramp completely? explain.
select the correct choice below, and fill in the answer box to complete your choice.
(simplify your answer. type an exact answer, using radicals as needed.)
a. yes. the length of the ramp is (square) feet. the rubber mat is equal to this length.
b. no. the length of the ramp is (square) feet. the rubber mat will be too short.
c. yes. the length of the ramp is (square) feet. the rubber mat is longer than this length.

Explanation:

Step1: Define variables for ramp sides

Let horizontal distance $x = 13k$, height $y = k$, ramp length $L$ (hypotenuse).

Step2: Apply Pythagorean theorem

$$L = \sqrt{(13k)^2 + k^2}$$

Step3: Simplify the expression

$$L = \sqrt{169k^2 + k^2} = \sqrt{170k^2} = k\sqrt{170}$$

Step4: Relate to given horizontal length

Given horizontal distance is 13 ft, so $13k = 13 \implies k=1$. Substitute $k=1$:
$$L = \sqrt{170} \approx 13.04$$

Step5: Compare to mat length

Mat length = 13 ft, $13 < \sqrt{170}$.

Answer:

B. No. The length of the ramp is $\sqrt{170}$ feet. The rubber mat will be too short.