Solved on Dec 16, 2023

Find the height of an aircraft with atmospheric pressure p=259p=259 mm Hg using the formula p=760e0.145hp=760 e^{-0.145 \mathrm{h}}. Round the height to two decimal places.

STEP 1

Assumptions
1. The atmospheric pressure pp is related to the height hh by the formula p=760e0.145hp = 760 e^{-0.145h}.
2. The pressure pp is measured in millimeters of mercury.
3. The height hh is measured in kilometers above sea level.
4. We are solving for two different pressures: 259 mmHg for an aircraft and 506 mmHg for a mountain.

STEP 2

For part (a), we are given that the atmospheric pressure pp is 259 mmHg. We need to solve for the height hh using the formula:
p=760e0.145hp = 760 e^{-0.145h}

STEP 3

Substitute 259 mmHg for pp in the formula to solve for hh:
259=760e0.145h259 = 760 e^{-0.145h}

STEP 4

Divide both sides of the equation by 760 to isolate the exponential term:
259760=e0.145h\frac{259}{760} = e^{-0.145h}

STEP 5

Calculate the left-hand side of the equation to simplify:
2597600.3408\frac{259}{760} \approx 0.3408

STEP 6

Take the natural logarithm of both sides to solve for hh:
ln(0.3408)=ln(e0.145h)\ln(0.3408) = \ln(e^{-0.145h})

STEP 7

Use the property of logarithms that ln(ex)=x\ln(e^x) = x to simplify the right-hand side:
ln(0.3408)=0.145h\ln(0.3408) = -0.145h

STEP 8

Divide both sides by -0.145 to solve for hh:
h=ln(0.3408)0.145h = \frac{\ln(0.3408)}{-0.145}

STEP 9

Calculate the value of hh:
hln(0.3408)0.145h \approx \frac{\ln(0.3408)}{-0.145}

STEP 10

Use a calculator to find the numerical value of hh:
hln(0.3408)0.1457.68h \approx \frac{\ln(0.3408)}{-0.145} \approx 7.68

STEP 11

Round hh to two decimal places as required:
h7.68 kilometersh \approx 7.68 \text{ kilometers}

STEP 12

For part (b), we are given that the atmospheric pressure pp is 506 mmHg. We need to solve for the height hh using the same formula:
p=760e0.145hp = 760 e^{-0.145h}

STEP 13

Substitute 506 mmHg for pp in the formula to solve for hh:
506=760e0.145h506 = 760 e^{-0.145h}

STEP 14

Divide both sides of the equation by 760 to isolate the exponential term:
506760=e0.145h\frac{506}{760} = e^{-0.145h}

STEP 15

Calculate the left-hand side of the equation to simplify:
5067600.6658\frac{506}{760} \approx 0.6658

STEP 16

Take the natural logarithm of both sides to solve for hh:
ln(0.6658)=ln(e0.145h)\ln(0.6658) = \ln(e^{-0.145h})

STEP 17

Use the property of logarithms that ln(ex)=x\ln(e^x) = x to simplify the right-hand side:
ln(0.6658)=0.145h\ln(0.6658) = -0.145h

STEP 18

Divide both sides by -0.145 to solve for hh:
h=ln(0.6658)0.145h = \frac{\ln(0.6658)}{-0.145}

STEP 19

Calculate the value of hh:
hln(0.6658)0.145h \approx \frac{\ln(0.6658)}{-0.145}

STEP 20

Use a calculator to find the numerical value of hh:
hln(0.6658)0.1452.41h \approx \frac{\ln(0.6658)}{-0.145} \approx 2.41

STEP 21

Round hh to two decimal places as required:
h2.41 kilometersh \approx 2.41 \text{ kilometers}
The height of the aircraft is approximately 7.68 kilometers, and the height of the mountain is approximately 2.41 kilometers.

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