If the temperature is constant, then the atmospheric pressure P (in pounds/square inch) varies with the altitude above sea level h in accordance with the law where is the atmospheric pressure at sea level and k is a constant. If the atmospheric pressure is at sea level and at 5,000 ft, how fast is the atmospheric pressure changing with respect to altitude at an altitude of 15,000 ft? Give the answer to four decimal places.

Question

If the temperature is constant, then the atmospheric pressure P (in pounds/square inch) varies with the altitude above sea level h in accordance with the law  where  is the atmospheric pressure at sea level and k is a constant. If the atmospheric pressure is  at sea level and  at 5,000 ft, how fast is the atmospheric pressure changing with respect to altitude at an altitude of 15,000 ft? Give the answer to four decimal places.

Quizplus · Accepted Answer

From the given information, we have:

P(h) = P_0 * e^(-k*h)

where P_0 = 11eaa8b1_dad8_2f15_b1b6_dfae5b686f21_TB6026_11 and P(5000) = 11eaa8b1_dad8_2f16_b1b6_9d820cce9a40_TB6026_11

Taking the derivative of P(h) with respect to h, we get:

dP/dh = -k * P_0 * e^(-k*h)

At h = 15000, we have:

dP/dh = -k * P_0 * e^(-k*15000)

To find the value of k, we can use the information that P(5000) = 11eaa8b1_dad8_2f16_b1b6_9d820cce9a40_TB6026_11:

11eaa8b1_dad8_2f16_b1b6_9d820cce9a40_TB6026_11 = P_0 * e^(-k*5000)

e^(-k*5000) = 11eaa8b1_dad8_2f16_b1b6_9d820cce9a40_TB6026_11 / P_0

-k*5000 = ln(11eaa8b1_dad8_2f16_b1b6_9d820cce9a40_TB6026_11 / P_0)

k = -ln(11eaa8b1_dad8_2f16_b1b6_9d820cce9a40_TB6026_11 / P_0) / 5000

Now we can substitute the values of P_0, k, and h into the expression for dP/dh:

dP/dh = -k * P_0 * e^(-k*h)

dP/dh = -(-ln(11eaa8b1_dad8_2f16_b1b6_9d820cce9a40_TB6026_11 / 11eaa8b1_dad8_2f15_b1b6_dfae5b686f21_TB6026_11) / 5000) * 11eaa8b1_dad8_2f15_b1b6_dfae5b686f21_TB6026_11 * e^(-(-ln(11eaa8b1_dad8_2f16_b1b6_9d820cce9a40_TB6026_11 / 11eaa8b1_dad8_2f15_b1b6_dfae5b686f21_TB6026_11) / 5000) * 15000)

dP/dh ≈ 11eaa8b1_dad8_5628_b1b6_6dc30241bec2_TB6026_11

If the Temperature Is Constant, Then the Atmospheric Pressure P