Question

a 15 - ft ladder leans against a wall so that the top reaches 7 ft above the ground.
round all answers to the nearest tenth.
what angle does the ladder make with the ground?
degrees
what angle does the ladder make with the wall?
degrees
how far is the base of the ladder from the bottom of the wall?
feet

Explanation:

Step1: Find the angle with the ground

We know the opposite side ($y = 7$ ft) and the hypotenuse ($h=15$ ft) of a right - triangle (ladder - wall - ground). Using the sine function $\sin\theta=\frac{opposite}{hypotenuse}$. Let $\theta$ be the angle with the ground. So $\sin\theta=\frac{7}{15}$. Then $\theta=\sin^{- 1}(\frac{7}{15})$.
$$\theta=\sin^{-1}(\frac{7}{15})\approx27.9^{\circ}$$

Step2: Find the angle with the wall

Let $\alpha$ be the angle with the wall. Since the sum of angles in a right - triangle is $90^{\circ}$, $\alpha = 90^{\circ}-\theta$. Substitute $\theta\approx27.9^{\circ}$, so $\alpha\approx90 - 27.9=62.1^{\circ}$

Step3: Find the distance from the base of the ladder to the wall

Using the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $c = 15$ (hypotenuse) and $b = 7$ (one leg). Let $a$ be the distance from the base of the ladder to the wall. Then $a=\sqrt{c^{2}-b^{2}}=\sqrt{15^{2}-7^{2}}=\sqrt{225 - 49}=\sqrt{176}\approx13.3$

Answer:

The angle with the ground is approximately $27.9$ degrees.
The angle with the wall is approximately $62.1$ degrees.
The distance from the base of the ladder to the wall is approximately $13.3$ feet.