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
66 feet
66 feet
55 feet
55 feet
680 inches
680 inches
240 inches
(1 point)
Responses
66 feet
66 feet
55 feet
55 feet
680 inches
680 inches
240 inches
Answers
GPT-4o mini
To find the height of the tree using trigonometric ratios, we can set up the problem using the information given:
1. Shaina's height: 5 feet 6 inches = 5 + 6/12 = 5.5 feet.
2. Distance from Shaina to the base of the tree: 20 feet.
3. Angle of elevation: 68 degrees.
We need to find the vertical height of the tree from the ground to the top, which includes both the height Shaina sees (from her eye level) and her height itself.
Let \( h \) be the height from Shaina's eye level to the top of the tree. We can use the tangent function, which relates the angle of elevation to the opposite side (the height we're looking for) and the adjacent side (the distance from Shaina to the tree):
\[
\tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}}
\]
Substituting the known values:
\[
\tan(68^\circ) = \frac{h}{20}
\]
Now, we can solve for \( h \):
\[
h = 20 \cdot \tan(68^\circ)
\]
Calculating \( \tan(68^\circ) \):
\[
\tan(68^\circ) \approx 2.4751
\]
Now, plug this value into the equation:
\[
h \approx 20 \cdot 2.4751 \approx 49.502 \text{ feet}
\]
Now, we add Shaina's eye level height to get the total height of the tree:
\[
\text{Total height of the tree} = h + 5.5 \text{ feet} \approx 49.502 + 5.5 \approx 55.002 \text{ feet}
\]
Rounding to the nearest feet, the height of the tree is approximately 55 feet.
Thus, the answer is:
**55 feet**
1. Shaina's height: 5 feet 6 inches = 5 + 6/12 = 5.5 feet.
2. Distance from Shaina to the base of the tree: 20 feet.
3. Angle of elevation: 68 degrees.
We need to find the vertical height of the tree from the ground to the top, which includes both the height Shaina sees (from her eye level) and her height itself.
Let \( h \) be the height from Shaina's eye level to the top of the tree. We can use the tangent function, which relates the angle of elevation to the opposite side (the height we're looking for) and the adjacent side (the distance from Shaina to the tree):
\[
\tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}}
\]
Substituting the known values:
\[
\tan(68^\circ) = \frac{h}{20}
\]
Now, we can solve for \( h \):
\[
h = 20 \cdot \tan(68^\circ)
\]
Calculating \( \tan(68^\circ) \):
\[
\tan(68^\circ) \approx 2.4751
\]
Now, plug this value into the equation:
\[
h \approx 20 \cdot 2.4751 \approx 49.502 \text{ feet}
\]
Now, we add Shaina's eye level height to get the total height of the tree:
\[
\text{Total height of the tree} = h + 5.5 \text{ feet} \approx 49.502 + 5.5 \approx 55.002 \text{ feet}
\]
Rounding to the nearest feet, the height of the tree is approximately 55 feet.
Thus, the answer is:
**55 feet**