Question

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

Explanation:

Step1: Recall slope formula

A 3 - percent slope means $\tan\theta=0.03$, where $\theta$ is the slope angle. Let the measured distance along the slope be $L = 210.23$ ft and the horizontal distance be $x$. We know that $\cos\theta=\frac{x}{L}$. Also, since $\tan\theta = 0.03=\frac{\sin\theta}{\cos\theta}$ and $\sin^{2}\theta+\cos^{2}\theta = 1$, we can find $\cos\theta$. From $\tan\theta=0.03$, we have $\sin\theta = 0.03\cos\theta$. Substituting into $\sin^{2}\theta+\cos^{2}\theta = 1$ gives $(0.03\cos\theta)^{2}+\cos^{2}\theta=1$.

Step2: Solve for $\cos\theta$

Expanding, we get $0.0009\cos^{2}\theta+\cos^{2}\theta=1$, so $1.0009\cos^{2}\theta = 1$. Then $\cos\theta=\sqrt{\frac{1}{1.0009}}\approx0.99955$.

Step3: Calculate horizontal distance

Using the formula $x = L\cos\theta$, substituting $L = 210.23$ ft and $\cos\theta\approx0.99955$, we have $x=210.23\times0.99955\approx210.13$ ft.

Answer:

$210.13$ ft