Question

1. a ski jump rises 3 feet over a run of 7 feet. a) what is the length of the surface of the jump? draw a picture to help. b) what is the angle of elevation of the jump?

Explanation:

Step1: Identify the right - triangle

The ski - jump forms a right - triangle where the vertical side (rise) $a = 3$ feet and the horizontal side (run) $b = 7$ feet. We want to find the hypotenuse $c$ (length of the surface of the jump) and the angle of elevation $\theta$.

Step2: Use the Pythagorean theorem for part a

The Pythagorean theorem is $c=\sqrt{a^{2}+b^{2}}$. Substitute $a = 3$ and $b = 7$ into the formula:
\[

$$\begin{align*} c&=\sqrt{3^{2}+7^{2}}\\ &=\sqrt{9 + 49}\\ &=\sqrt{58}\approx7.62\text{ feet} \end{align*}$$

\]

Step3: Use the tangent function for part b

The tangent of the angle of elevation $\theta$ in a right - triangle is given by $\tan\theta=\frac{a}{b}$. Here, $\tan\theta=\frac{3}{7}$. Then $\theta=\arctan(\frac{3}{7})$.
\[

$$\begin{align*} \theta&=\arctan(\frac{3}{7})\\ &\approx\arctan(0.4286)\\ &\approx23.2^{\circ} \end{align*}$$

\]

Answer:

a) The length of the surface of the jump is approximately $\sqrt{58}\approx7.62$ feet.
b) The angle of elevation of the jump is approximately $\arctan(\frac{3}{7})\approx23.2^{\circ}$.