Question

You are to construct a decorative wood shelf that has two metal angle supports that fasten to the bottom of the shelf and resemble a right triangle. The shelves are to stick out from the wall 1 foot, and the diagonal distance of the metal piece is 1.25 feet. Using the inverse of sine, what is the approximate angle formed where the metal piece rests on the wall? (1 point)
0610-
• 53°
037°
• 90°

Answer 1

To find the angle formed where the metal piece rests on the wall, we can use the properties of a right triangle.

Given:
- The horizontal distance from the wall to the outer edge of the shelf (one leg of the triangle) = 1 foot
- The diagonal distance of the metal piece (the hypotenuse of the triangle) = 1.25 feet

We can use the sine function, which in a right triangle is defined as:

\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]

In this case, the opposite side to the angle \(\theta\) (the angle formed where the metal piece rests on the wall) is the vertical distance from the shelf to where the angle support meets the wall. We can determine this length using the Pythagorean theorem:

\[
a^2 + b^2 = c^2
\]

where:
- \( a = 1 \, \text{foot} \) (horizontal leg),
- \( b \) is the vertical leg,
- \( c = 1.25 \, \text{feet} \) (hypotenuse).

Let's calculate \( b \):

\[
1^2 + b^2 = (1.25)^2
\]
\[
1 + b^2 = 1.5625
\]
\[
b^2 = 1.5625 - 1
\]
\[
b^2 = 0.5625
\]
\[
b = \sqrt{0.5625} \approx 0.75 \, \text{feet}
\]

Now we can use the sine inverse function to find the angle \( \theta \):

\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{b}{c} = \frac{0.75}{1.25}
\]

Calculating this gives:

\[
\sin(\theta) = 0.6
\]

Now we take the inverse sine (arcsin):

\[
\theta = \arcsin(0.6) \approx 36.87^\circ
\]

Since this is approximately \( 37^\circ \), the closest answer choice is:

**37°**.