Looking up, Shen sees two hot air balloons in the sky as shown. He determines that the lower hot air balloon is 425 meters away, at an angle of from the
vertical. The higher hot air balloon is 560 meters away, at an angle of from the vertical. How much higher is the balloon on the right than the balloon on the
left?
Do not round any intermediate computations. Round your answer to the nearest tenth.
Note that the figure below is not drawn to scale.

Ask by Schmidt Barker. in the United States

Mar 29,2025

Question

Looking up, Shen sees two hot air balloons in the sky as shown. He determines that the lower hot air balloon is 425 meters away, at an angle of from the
vertical. The higher hot air balloon is 560 meters away, at an angle of from the vertical. How much higher is the balloon on the right than the balloon on the
left?
Do not round any intermediate computations. Round your answer to the nearest tenth.
Note that the figure below is not drawn to scale.

Ask by Schmidt Barker. in the United States

Mar 29,2025

UpStudy AI · Accepted Answer

We first note that the measured distance to each balloon is along the line of sight, and the angle provided is with respect to the vertical. Thus the vertical component (height) from the observer’s position is given by

$$
h = d \cos \theta,
$$

where $$d$$ is the given distance and $$\theta$$ is the angle from the vertical.

For the left (lower) balloon we have
$$
h_1 = 425 \cos(31^\circ).
$$
For the right (higher) balloon we have
$$
h_2 = 560 \cos(43^\circ).
$$

The difference in height is then
$$
\Delta h = h_2 - h_1 = 560 \cos(43^\circ) - 425 \cos(31^\circ).
$$

Do not round any intermediate computations. Now, substituting the values:

$$
\text{Using exact computations:}
$$
$$
h_1 = 425 \cos(31^\circ), \quad h_2 = 560 \cos(43^\circ).
$$

Calculating (using a calculator in its full precision):

$$
\cos(31^\circ) \approx 0.8571673007,
$$
$$
\cos(43^\circ) \approx 0.7313537016.
$$

Then,
$$
h_1 \approx 425 \times 0.8571673007 \approx 364.4291778,
$$
$$
h_2 \approx 560 \times 0.7313537016 \approx 409.9580730.
$$

So the difference in height is
$$
\Delta h \approx 409.9580730 - 364.4291778 \approx 45.5288952.
$$

Rounding to the nearest tenth, the balloon on the right is

$$
\boxed{45.5 \text{ meters}}
$$

higher than the balloon on the left.
The balloon on the right is 45.5 meters higher than the balloon on the left.

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