Question
page 15 of 15 Solving Right Triangles Quick Check 1 of 51 of 5 Items Question Use the image to answer the question. Use trigonometric ratios to solve the following problem. Shaina, who is 5 feet, 6 inches tall, is standing 20 feet from the base of a tree when she sees a pileated woodpecker at the top of the tree. The bird is at an angle of elevation of 68 degrees above Shaina’s line of sight. What is the height of the tree? (1 point) Responses 680 inches 680 inches 55 feet 55 feet 240 inches 240 inches 66 feet 66 feet Skip to navigation
Answers
GPT-4o mini
To find the height of the tree that Shaina is looking at, we can use the information given and apply trigonometric ratios.
First, let's convert Shaina's height from feet and inches to just feet. Shaina is 5 feet 6 inches tall, which is:
\[
5 \text{ feet} + \frac{6 \text{ inches}}{12} = 5 + 0.5 = 5.5 \text{ feet}
\]
Now, let’s denote:
- \( h \) = height of the tree,
- \( d \) = distance from Shaina to the base of the tree = 20 feet,
- \( \theta \) = angle of elevation = 68 degrees.
When Shaina looks up at the bird atop the tree, the total height from the ground to the bird is the height of the tree \( h \) minus Shaina's height (5.5 feet). The angle of elevation (68 degrees) gives us a way to relate these heights using the tangent function.
Using the tangent ratio:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h - 5.5}{20}
\]
Substituting the angle and rearranging gives:
\[
\tan(68^\circ) = \frac{h - 5.5}{20}
\]
Calculating \( \tan(68^\circ) \):
\[
\tan(68^\circ) \approx 2.4751 \quad \text{(using a calculator)}
\]
Now plug this value into the equation:
\[
2.4751 = \frac{h - 5.5}{20}
\]
Multiply both sides by 20:
\[
20 \cdot 2.4751 = h - 5.5
\]
\[
49.502 = h - 5.5
\]
Now add 5.5 to both sides to solve for \( h \):
\[
h = 49.502 + 5.5 \approx 55.002
\]
Therefore, the height of the tree is approximately 55.002 feet, which rounds to:
\[
\text{Height of the tree} \approx 55 \text{ feet}
\]
So the answer is **55 feet**.
First, let's convert Shaina's height from feet and inches to just feet. Shaina is 5 feet 6 inches tall, which is:
\[
5 \text{ feet} + \frac{6 \text{ inches}}{12} = 5 + 0.5 = 5.5 \text{ feet}
\]
Now, let’s denote:
- \( h \) = height of the tree,
- \( d \) = distance from Shaina to the base of the tree = 20 feet,
- \( \theta \) = angle of elevation = 68 degrees.
When Shaina looks up at the bird atop the tree, the total height from the ground to the bird is the height of the tree \( h \) minus Shaina's height (5.5 feet). The angle of elevation (68 degrees) gives us a way to relate these heights using the tangent function.
Using the tangent ratio:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h - 5.5}{20}
\]
Substituting the angle and rearranging gives:
\[
\tan(68^\circ) = \frac{h - 5.5}{20}
\]
Calculating \( \tan(68^\circ) \):
\[
\tan(68^\circ) \approx 2.4751 \quad \text{(using a calculator)}
\]
Now plug this value into the equation:
\[
2.4751 = \frac{h - 5.5}{20}
\]
Multiply both sides by 20:
\[
20 \cdot 2.4751 = h - 5.5
\]
\[
49.502 = h - 5.5
\]
Now add 5.5 to both sides to solve for \( h \):
\[
h = 49.502 + 5.5 \approx 55.002
\]
Therefore, the height of the tree is approximately 55.002 feet, which rounds to:
\[
\text{Height of the tree} \approx 55 \text{ feet}
\]
So the answer is **55 feet**.