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Sagot :
Sure, let's go through this problem step-by-step to find the height of the taller building.
### Step 1: Understand the problem
We are given:
- The height of the first building is 16 meters.
- The distance between the two buildings is 30 meters.
- The angle of elevation from the top of the first building to the top of the second building is 48°.
We need to determine the height of the taller building.
### Step 2: Visualize the problem
Imagine the scenario as a right triangle where:
- One leg of the triangle represents the distance between the two buildings (30 meters).
- The other leg of the triangle represents the height difference between the top of the two buildings.
- The angle between the base and the hypotenuse (from the top of the first building to the top of the second building) is 48°.
### Step 3: Use trigonometric relations
We can use the tangent function, which is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. Here, the opposite side is the height difference between the buildings, and the adjacent side is the distance between the buildings.
[tex]\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \][/tex]
Given:
- [tex]\(\theta = 48^\circ\)[/tex]
- Adjacent side = 30 meters
So, the equation becomes:
[tex]\[ \tan(48^\circ) = \frac{\text{height difference}}{30} \][/tex]
### Step 4: Solve for height difference
Rearrange the equation to solve for the height difference:
[tex]\[ \text{height difference} = 30 \times \tan(48^\circ) \][/tex]
### Step 5: Calculate the height difference
From the problem, we know these calculations yield:
[tex]\[ \tan(48^\circ) \approx 1.1106 \][/tex] (using a calculator or trigonometric table)
So:
[tex]\[ \text{height difference} = 30 \times 1.1106 \approx 33.31837544487579 \ \text{meters} \][/tex]
### Step 6: Find the height of the taller building
The height of the taller building is the height of the first building plus the height difference:
[tex]\[ \text{Height of taller building} = \text{Height of first building} + \text{Height difference} \][/tex]
Given the first building is 16 meters tall:
[tex]\[ \text{Height of taller building} = 16 + 33.31837544487579 \][/tex]
### Step 7: Final calculation
Adding these together gives:
[tex]\[ \text{Height of taller building} \approx 49.31837544487579 \ \text{meters} \][/tex]
Thus, the height of the taller building is approximately 49.32 meters.
### Step 1: Understand the problem
We are given:
- The height of the first building is 16 meters.
- The distance between the two buildings is 30 meters.
- The angle of elevation from the top of the first building to the top of the second building is 48°.
We need to determine the height of the taller building.
### Step 2: Visualize the problem
Imagine the scenario as a right triangle where:
- One leg of the triangle represents the distance between the two buildings (30 meters).
- The other leg of the triangle represents the height difference between the top of the two buildings.
- The angle between the base and the hypotenuse (from the top of the first building to the top of the second building) is 48°.
### Step 3: Use trigonometric relations
We can use the tangent function, which is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. Here, the opposite side is the height difference between the buildings, and the adjacent side is the distance between the buildings.
[tex]\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \][/tex]
Given:
- [tex]\(\theta = 48^\circ\)[/tex]
- Adjacent side = 30 meters
So, the equation becomes:
[tex]\[ \tan(48^\circ) = \frac{\text{height difference}}{30} \][/tex]
### Step 4: Solve for height difference
Rearrange the equation to solve for the height difference:
[tex]\[ \text{height difference} = 30 \times \tan(48^\circ) \][/tex]
### Step 5: Calculate the height difference
From the problem, we know these calculations yield:
[tex]\[ \tan(48^\circ) \approx 1.1106 \][/tex] (using a calculator or trigonometric table)
So:
[tex]\[ \text{height difference} = 30 \times 1.1106 \approx 33.31837544487579 \ \text{meters} \][/tex]
### Step 6: Find the height of the taller building
The height of the taller building is the height of the first building plus the height difference:
[tex]\[ \text{Height of taller building} = \text{Height of first building} + \text{Height difference} \][/tex]
Given the first building is 16 meters tall:
[tex]\[ \text{Height of taller building} = 16 + 33.31837544487579 \][/tex]
### Step 7: Final calculation
Adding these together gives:
[tex]\[ \text{Height of taller building} \approx 49.31837544487579 \ \text{meters} \][/tex]
Thus, the height of the taller building is approximately 49.32 meters.
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