Problem 36 - Algebra and Trigonometry Real Mathematics, Real People

Question

The height of an outdoor basketball backboard is \(12 \frac{1}{2}\) feet, and the backboard casts a shadow \(17 \frac{1}{3}\) feet long. (a) Draw a right triangle that gives a visual representation of the problem. Label the known and unknown quantities. (b) Use a trigonometric function to write an equation involving the unknown angle of elevation. (c) Find the angle of elevation of the sun.

Step-by-Step Solution

Answer

The angle of elevation of the sun is obtained by evaluating \(\arctan\left(\frac{12 \frac{1}{2}}{17 \frac{1}{3}}\right)\). This will yield the answer in degrees when using calculator in degree mode.

Step 1: Drawing a Right Triangle

Draw a right triangle, labelling the vertical line as \(12 \frac{1}{2}\) feet (height of the basketball backboard), the horizontal line as \(17 \frac{1}{3}\) feet (length of the shadow), and the angle between the two lines as the unknown angle of elevation E.

Step 2: Writing a Trigonometric Equation

In the drawn triangle, the height of the backboard represents the opposite side, and the length of shadow represents the adjacent side concerning angle E. We can use a trigonometric function that uses these sides, i.e., tangent. So, formulate the equation as: \(\tan(E) = \frac{\text{{height}}}{\text{{shadow length}}} = \frac{12 \frac{1}{2}}{17 \frac{1}{3}}\)

Step 3: Solving the Equation

Now we solve the equation for E, so \(E = \arctan\left(\frac{12 \frac{1}{2}}{17 \frac{1}{3}}\right)\). Use a calculator to get the answer, ensure to convert the decimal to degrees.

Step 4: Making a Conclusion

From step 3, we obtain the angle in degrees. That's the angle of elevation of the sun.