Question

1. a distance of 210.23 ft was measured along a 3 percent slope. compute the horizontal distance.

Explanation:

Step1: Recall slope - distance relationship

Let the measured distance along the slope be $L = 210.23$ ft and the slope $s=0.03$. If the horizontal distance is $x$ and the vertical distance is $y$, the slope $s=\frac{y}{x}$ and by the Pythagorean theorem $L^{2}=x^{2}+y^{2}$. Since $y = sx$, we have $L^{2}=x^{2}+(sx)^{2}=x^{2}(1 + s^{2})$.

Step2: Solve for $x$

We can rewrite the equation $L^{2}=x^{2}(1 + s^{2})$ as $x=\frac{L}{\sqrt{1 + s^{2}}}$. Substitute $L = 210.23$ ft and $s = 0.03$ into the formula. First, calculate $1 + s^{2}=1+(0.03)^{2}=1 + 0.0009=1.0009$. Then $\sqrt{1 + s^{2}}=\sqrt{1.0009}\approx1.00045$. And $x=\frac{210.23}{1.00045}\approx210.13$ ft.

Answer:

$210.13$ ft