Question

a. a wheelchair ramp with a length of 137 inches has a horizontal distance of 105 inches. what is the ramps vertical distance?
b. construction laws are very specific when it comes to access ramps for the disabled. every vertical rise of 1 inch requires a horizontal run of 12 inches. does this ramp satisfy the requirements?
a. the vertical distance of the ramp is 88 inches.
b. does this ramp satisfy the requirements? choose the correct answer below.
a. yes, the ramp does satisfy the requirements because, using the given dimensions, the ratio of vertical rise to horizontal run is greater than 1:12.
b. no, the ramp does not satisfy the requirements because, using the given dimensions, the ratio of vertical rise to horizontal run is less than 12:1.
c. no, the ramp does not satisfy the requirements because, using the given dimensions, the ratio of vertical rise to horizontal run does not equal 1:12.
d. yes, the ramp does satisfy the requirements because, using the given dimensions, the ratio of vertical rise to horizontal run is equal to 1:12.

Explanation:

Step1: Use Pythagorean theorem for part a

Let the vertical distance be $x$. According to the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $c = 137$ (hypotenuse - length of ramp) and $b = 105$ (horizontal distance). So $x=\sqrt{137^{2}-105^{2}}$.
\[

$$\begin{align*} x&=\sqrt{(137 + 105)(137- 105)}\\ &=\sqrt{242\times32}\\ &=\sqrt{7744}\\ & = 88 \end{align*}$$

\]

Step2: Calculate the ratio for part b

The vertical rise is $x = 88$ inches and the horizontal run is $105$ inches. The ratio of vertical rise to horizontal run is $\frac{88}{105}\approx0.84$. The required ratio is $\frac{1}{12}\approx0.083$. Since $\frac{88}{105}>\frac{1}{12}$, the ramp satisfies the requirements.

Answer:

a. 88 inches
b. A. Yes, the ramp does satisfy the requirements because, using the given dimensions, the ratio of vertical rise to horizontal run is greater than 1:12.