A building in a downtown business area casts a shadow that measures 88 meters along the ground. The straight-line distance from the top of the building to the end of the shadow it creates is at a 32° angle with the ground. What is the approximate height of the building? Round your answer to the nearest meter.
how many meters is the building?

This problem can be solved using trigonometry, specifically the sine function. Here's how:

### Step 1: Identify the right triangle
- The shadow along the ground (88 meters) forms the base of a right triangle.
- The straight-line distance from the top of the building to the end of the shadow forms the hypotenuse of the triangle.
- The angle between the hypotenuse and the ground is 32°.

### Step 2: Use the sine function
The sine of an angle in a right triangle is given by:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
Where:
- \(\theta = 32^\circ\) (the angle),
- The opposite side is the height of the building (which we are solving for),
- The hypotenuse is the straight-line distance.

Let \( h \) represent the height of the building, and \( d \) represent the hypotenuse. We don't have \( d \) directly, but we can find it using the cosine function and the length of the shadow.

### Step 3: Find the hypotenuse (distance)
Use the cosine function to find the hypotenuse \( d \):
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{88}{d}
\]
\[
d = \frac{88}{\cos(32^\circ)}
\]

### Step 4: Solve for \( d \)
Now we compute \( d \):
\[
d \approx \frac{88}{\cos(32^\circ)} \approx \frac{88}{0.848} \approx 103.77 \text{ meters}
\]

### Step 5: Use the sine function to find the height
Now that we know \( d \approx 103.77 \), we can use the sine function to find the height:
\[
\sin(32^\circ) = \frac{h}{103.77}
\]
\[
h = 103.77 \times \sin(32^\circ)
\]
\[
h \approx 103.77 \times 0.530 \approx 55.0 \text{ meters}
\]

### Final Answer:
The height of the building is approximately **55 meters**.

Would you like more details on any step or have any questions?

### Related Questions:
1. How does the cosine function relate to right triangles in trigonometry?
2. What other functions could be used to solve for height in different scenarios?
3. What if the angle was smaller, how would that affect the height?
4. How would the calculation change if the shadow was on a slope instead of flat ground?
5. How can the Pythagorean theorem be used as an alternative in this problem?

### Tip:
When solving right triangle problems, always start by identifying which sides and angles you know, then choose the appropriate trigonometric function based on that information.

A building in a downtown business area casts a shadow that measures 88 meters along the ground. The straight-line distance from the top of the building to the end of the shadow it creates is at a 32° angle with the ground. What is the approximate height of the building? Round your answer to the nearest meter.

Learn how to calculate the height of a building using trigonometry. A shadow of 88 meters and an angle of 32° form the basis for solving this problem. Key concepts include the sine and cosine functions in right triangles, perfect for Grades 9-12.

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