Solved on Jan 29, 2024

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

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

STEP 1

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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The two airplanes will be at the same altitude in 464003200+2600\sqrt{\frac{46400}{3200 + 2600}}3200+260046400​​ minutes, when their altitude will be 232002320023200 feet.

STEP 1

STEP 2

Let's denote the time at which the two airplanes will be at the same altitude as ttt (in minutes).

STEP 3

The altitude of the descending airplane after ttt minutes can be described by the following equation:Altitudedescending=46,400−3,200tAltitude_{descending} = 46,400 - 3,200tAltitudedescending​=46,400−3,200t

STEP 4

The altitude of the ascending airplane after ttt minutes can be described by the following equation:Altitudeascending=2,600tAltitude_{ascending} = 2,600tAltitudeascending​=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=2,600t46,400 - 3,200t = 2,600t46,400−3,200t=2,600t

STEP 6

Combine like terms by adding 3,200t3,200t3,200t to both sides of the equation:46,400=3,200t+2,600t46,400 = 3,200t + 2,600t46,400=3,200t+2,600t

STEP 7

Combine the terms on the right side of the equation:46,400=5,800t46,400 = 5,800t46,400=5,800t

STEP 8

Solve for ttt by dividing both sides of the equation by 5,8005,8005,800:t=46,4005,800t = \frac{46,400}{5,800}t=5,80046,400​

STEP 9

Calculate the value of ttt:t=46,4005,800=8t = \frac{46,400}{5,800} = 8t=5,80046,400​=8

STEP 10

STEP 11

Plug in the value of ttt into the altitude equation:Altitudeascending=2,600×8Altitude_{ascending} = 2,600 \times 8Altitudeascending​=2,600×8

STEP 12

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

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

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

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

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

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

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

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

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

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

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

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

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.