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Answer :
We begin with the two equations that give the altitude (in thousands of feet) of the two planes as functions of time (in minutes):
$$
\text{Plane A: } y = 12 - 2.5x
$$
$$
\text{Plane B: } y = 1 + 4x
$$
Since the planes will be at the same altitude when their $y$-values are equal, we set the equations equal to each other:
$$
12 - 2.5x = 1 + 4x
$$
**Step 1. Solve for $x$**
First, subtract $1$ from $12$:
$$
12 - 1 = 11
$$
Now, rewrite the equation by collecting the $x$ terms on one side:
$$
11 = 4x + 2.5x
$$
Combine like terms:
$$
11 = 6.5x
$$
Solve for $x$ by dividing both sides by $6.5$:
$$
x = \frac{11}{6.5} \approx 1.6923 \text{ minutes}
$$
**Step 2. Find the altitude where they meet**
It is convenient to use the equation for Plane B. Substitute $x \approx 1.6923$ into the equation for Plane B:
$$
y = 1 + 4x \approx 1 + 4(1.6923) \approx 1 + 6.7692 \approx 7.7692
$$
Since $y$ is given in thousands of feet, the altitude in feet is:
$$
\text{Altitude} \approx 7.7692 \times 1000 \approx 7769 \text{ feet}
$$
**Step 3. Convert the time into minutes and seconds**
We have $x \approx 1.6923$ minutes. The whole number part represents the minutes. So the time is 1 minute plus a fractional part:
- Minutes: $1$
- Fractional Part: $0.6923$ minutes
To convert the fractional part into seconds, multiply by 60:
$$
\text{Seconds} = 0.6923 \times 60 \approx 41.54 \text{ seconds}
$$
**Final Answer**
The two planes will be at the same altitude after approximately $1.69$ minutes (or $1$ minute and $41.5$ seconds). At that time, they will both be at an altitude of about $7769$ feet.
$$
\text{Plane A: } y = 12 - 2.5x
$$
$$
\text{Plane B: } y = 1 + 4x
$$
Since the planes will be at the same altitude when their $y$-values are equal, we set the equations equal to each other:
$$
12 - 2.5x = 1 + 4x
$$
**Step 1. Solve for $x$**
First, subtract $1$ from $12$:
$$
12 - 1 = 11
$$
Now, rewrite the equation by collecting the $x$ terms on one side:
$$
11 = 4x + 2.5x
$$
Combine like terms:
$$
11 = 6.5x
$$
Solve for $x$ by dividing both sides by $6.5$:
$$
x = \frac{11}{6.5} \approx 1.6923 \text{ minutes}
$$
**Step 2. Find the altitude where they meet**
It is convenient to use the equation for Plane B. Substitute $x \approx 1.6923$ into the equation for Plane B:
$$
y = 1 + 4x \approx 1 + 4(1.6923) \approx 1 + 6.7692 \approx 7.7692
$$
Since $y$ is given in thousands of feet, the altitude in feet is:
$$
\text{Altitude} \approx 7.7692 \times 1000 \approx 7769 \text{ feet}
$$
**Step 3. Convert the time into minutes and seconds**
We have $x \approx 1.6923$ minutes. The whole number part represents the minutes. So the time is 1 minute plus a fractional part:
- Minutes: $1$
- Fractional Part: $0.6923$ minutes
To convert the fractional part into seconds, multiply by 60:
$$
\text{Seconds} = 0.6923 \times 60 \approx 41.54 \text{ seconds}
$$
**Final Answer**
The two planes will be at the same altitude after approximately $1.69$ minutes (or $1$ minute and $41.5$ seconds). At that time, they will both be at an altitude of about $7769$ feet.
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