Question The two airplanes will be at the same altitude in $\sqrt{\frac{46400}{3200 + 2600}}$ minutes, when their altitude will be $23200$ feet.

Studdy Solution

STEP 1

Assumptions
1. The descending airplane starts at 46,400 feet above the ground.
2. The descending airplane descends at a rate of 3,200 feet per minute.
3. The ascending airplane starts at ground level (0 feet).
4. The ascending airplane ascends at a rate of 2,600 feet per minute.
5. We are looking for the time at which both airplanes will be at the same altitude.

STEP 2

Let's denote the time at which the two airplanes will be at the same altitude as $t$ (in minutes).

STEP 3

The altitude of the descending airplane after $t$ minutes can be described by the following equation:
$Altitude_{descending} = 46,400 - 3,200t$

STEP 4

The altitude of the ascending airplane after $t$ minutes can be described by the following equation:
$Altitude_{ascending} = 2,600t$

STEP 5

To find the time when both airplanes are at the same altitude, we set the altitudes equal to each other:
$46,400 - 3,200t = 2,600t$

STEP 6

Combine like terms by adding $3,200t$ to both sides of the equation:
$46,400 = 3,200t + 2,600t$

STEP 7

Combine the terms on the right side of the equation:
$46,400 = 5,800t$

STEP 8

Solve for $t$ by dividing both sides of the equation by $5,800$ :
$t = \frac{46,400}{5,800}$

STEP 9

Calculate the value of $t$ :
$t = \frac{46,400}{5,800} = 8$

STEP 10

Now that we have the time $t$ , we can find the altitude at which both airplanes will be at the same altitude. We can use either the descending or ascending airplane's altitude equation. Let's use the ascending airplane's altitude equation:
$Altitude_{ascending} = 2,600t$

STEP 11

Plug in the value of $t$ into the altitude equation:
$Altitude_{ascending} = 2,600 \times 8$

STEP 12

Calculate the altitude:
$Altitude_{ascending} = 2,600 \times 8 = 20,800$
The two airplanes will be at the same altitude in 8 minutes, when their altitude will be 20,800 feet.

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