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 that the ladder is the hypotenuse (\(c = 15\) ft) and the height on the wall is the opposite side (\(a=7\) ft) for the angle with the ground. Using the sine function \(\sin\theta=\frac{a}{c}\). So \(\sin\theta=\frac{7}{15}\). Then \(\theta=\sin^{- 1}(\frac{7}{15})\approx27.9^{\circ}\)

Step2: Find the angle with the wall

Since the two angles (with the ground and with the wall) are complementary (\(\theta_1+\theta_2 = 90^{\circ}\)). Let \(\theta_1\) be the angle with the ground (\(\theta_1\approx27.9^{\circ}\)), then the angle with the wall \(\theta_2=90^{\circ}-\theta_1\approx90 - 27.9=62.1^{\circ}\)

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

Using the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(a = 7\), \(c = 15\). We solve for \(b\): \(b=\sqrt{c^{2}-a^{2}}=\sqrt{15^{2}-7^{2}}=\sqrt{225 - 49}=\sqrt{176}\approx13.3\)

Answer:

27.9 degrees
62.1 degrees
13.3 feet