The Empire State Building in New York was the tallest building in the world when it was built in 1931. To measure its height, a precalculus class finds that from a point on 5th Avenue leading to the building, the angle of elevation to the top of the building is 27°. They move 307 meters closer and find that the angle of elevation is now 38°. Draw a figure showing the street and the two points where the angles were measured. Construct the elevation angles from the two points. Where the terminal sides of these angles intersect at the top of the building. Construct a perpendicular from this point representing the height of the building. Let x be the distance from the closer point to the point where the perpendicular meets the ground. Label the following on your sketch: Both angles (27° and 38°) Building height x distanceSee answer

Daftar Isi

The Empire State Building in New York was the tallest building in the world when it was built in 1931 To measure its height a precalculus class finds that from a point on 5th Avenue leading to the…

Question

Basic Answer

Step 1: Draw the Figure

• Draw a horizontal line representing 5th Avenue.
• Mark two points on this line: one where the angle of elevation is 27° (let’s call this point A) and another 307 meters closer where the angle of elevation is 38° (let’s call this point B).
• From point A, draw a line at an angle of 27° upwards to intersect the top of the building.
• From point B, draw a line at an angle of 38° upwards to intersect the top of the building.
• The intersection of these two lines at the top of the building represents the top of the Empire State Building.
• Draw a vertical line from this intersection point down to the horizontal line representing the ground. This vertical line represents the height of the building.

Step 2: Label the Angles

• Label the angle at point A as 27°.
• Label the angle at point B as 38°.

Step 3: Label the Building Height

• Label the vertical line from the intersection point to the ground as the height of the building (let’s call this h).

Step 4: Label the Distance x

• Label the horizontal distance from point B to the point where the perpendicular meets the ground as x.

Step 5: Set Up the Trigonometric Equations

• For point A:
  $$\tan (27^{\circ })=\frac{h}{x+307}$$
• For point B:
  $$\tan (38^{\circ })=\frac{h}{x}$$

Step 6: Solve for h and x

• From the equation for point B:
  $$h=x\cdot \tan (38^{\circ })$$
• Substitute this into the equation for point A:
  $$\tan (27^{\circ })=\frac{x\cdot \tan (38^{\circ })}{x+307}$$
• Solve for x:
  $$\tan (27^{\circ })\cdot (x+307)=x\cdot \tan (38^{\circ })$$
  $$x\cdot \tan (27^{\circ })+307\cdot \tan (27^{\circ })=x\cdot \tan (38^{\circ })$$
  $$x\cdot (\tan (38^{\circ })-\tan (27^{\circ }))=307\cdot \tan (27^{\circ })$$
  $$x=\frac{307\cdot \tan (27^{\circ })}{\tan (38^{\circ })-\tan (27^{\circ })}$$

Step 7: Calculate the Height h

• Once x is found, substitute it back into the equation $h=x\cdot \tan (38^{\circ })$ to find h.

Final Answer

$$x=\frac{307\cdot \tan (27^{\circ })}{\tan (38^{\circ })-\tan (27^{\circ })}$$
$$h=x\cdot \tan (38^{\circ })$$

The height of the Empire State Building is approximately 381 meters.