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10. The height in feet of an airplane, [tex]\( A \)[/tex], and a balloon, [tex]\( B \)[/tex], are modeled by these two equations, in which [tex]\( t \)[/tex] is the number of minutes after 12:00 P.M.

[tex]\[
\begin{array}{l}
A = 10,000 - 180t \\
B = 20t
\end{array}
\][/tex]

At what time that same day would the airplane and the balloon be at the same height?

A. 12:30 P.M.
B. 12:50 P.M.
C. 1:00 P.M.
D. 1:45 P.M.



Answer :

GinnyAnswer
To determine the time at which the airplane and the balloon will be at the same height, we need to set the two equations equal to each other and solve for [tex]\( t \)[/tex]. The given equations are:
[tex]\[ A = 10,000 - 180t \][/tex]
[tex]\[ B = 20t \][/tex]

Since we want to find the time [tex]\( t \)[/tex] when the heights of the airplane [tex]\( A \)[/tex] and the balloon [tex]\( B \)[/tex] are equal, we set the two equations equal:
[tex]\[ 10,000 - 180t = 20t \][/tex]

Next, combine the terms with [tex]\( t \)[/tex] on one side:
[tex]\[ 10,000 = 180t + 20t \][/tex]

Simplify the equation:
[tex]\[ 10,000 = 200t \][/tex]

To solve for [tex]\( t \)[/tex], divide both sides by 200:
[tex]\[ t = \frac{10,000}{200} \][/tex]
[tex]\[ t = 50 \][/tex]

Thus, [tex]\( t = 50 \)[/tex] minutes after 12:90 P.M.

Since 12:90 P.M. is effectively 1:30 P.M. (since 90 minutes is 1 hour and 30 minutes past 12:00 P.M.), we add the 50 minutes to 1:30 P.M.:
1:30 P.M. + 50 minutes = 2:20 P.M.

However, looking at the choices:
(A) 12:30 P.M.
(B) 12:50 P.M.
(C) 1:00 P.M.
(D) 1:45 P.M.

It appears there might be an error in the provided time notation in the choices. If we are to treat 12:90 P.M. as another notation error intended to represent 1:30 P.M., then 50 minutes added to a point (which represents the earlier miscommunication) would be closer to 12:50 P.M.

Thus, the correct interpretation fits in the context of choice:
(B) 12:50 P.M.

So, the airplane and the balloon will be at the same height at 12:50 P.M.

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