Question

a ladder leans against the wall of a building. the ladder measures 44 inches and forms an angle of 58° with the ground. how far from the ground, in inches, is the top of the ladder? how far from the wall, in inches, is the base of the ladder? round to two decimal places as needed. show your work here ground to top, in inches: base to wall, in inches:

Explanation:

Step1: Identify the triangle type

This is a right - triangle problem where the ladder is the hypotenuse ($c = 44$ inches), the angle with the ground $\theta=58^{\circ}$. Let the height from the ground to the top of the ladder be $a$ (opposite to $\theta$) and the distance from the base of the ladder to the wall be $b$ (adjacent to $\theta$).

Step2: Calculate the height (ground to top)

We use the sine function: $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{a}{c}$.
So, $a = c\times\sin\theta$.
Substitute $c = 44$ and $\theta = 58^{\circ}$:
$a=44\times\sin(58^{\circ})$.
$\sin(58^{\circ})\approx0.8480$, so $a = 44\times0.8480\approx37.31$ inches.

Step3: Calculate the distance (base to wall)

We use the cosine function: $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{b}{c}$.
So, $b = c\times\cos\theta$.
Substitute $c = 44$ and $\theta = 58^{\circ}$:
$\cos(58^{\circ})\approx0.5299$, so $b = 44\times0.5299\approx23.32$ inches.

Answer:

ground to top, in inches: $\boldsymbol{37.31}$
base to wall, in inches: $\boldsymbol{23.32}$