There is a mathematical relationship between an airplane’s weight x and the runway length required at takeoff. For certain types of airplanes, the minimum runway length L in thousands of feet may be modeled by L(x) = 1.3 ln x, where x is in thousands of pounds. a) Estimate the runway length needed for an airplane weighing 10,000 pounds. (b) Does a 20,000-pound airplane need twice the runway length that a 10,000-pound airplane needs? Explain

Question

There is a mathematical relationship between an airplane’s weight x and the runway length required at takeoff. For certain types of airplanes, the minimum runway length L in thousands of feet may be modeled by L(x) = 1.3 ln x, where x is in thousands of pounds. a) Estimate the runway length needed for an airplane weighing 10,000 pounds. (b) Does a 20,000-pound airplane need twice the runway length that a 10,000-pound airplane needs? Explain

Answer 1

To estimate the runway length needed for an airplane weighing 10,000 pounds, we can use the given mathematical model:

L(x) = 1.3 ln x

Substitute x = 10,000 into the equation:

L(10) = 1.3 ln 10,000

Now, let's find the natural logarithm of 10,000 using a calculator or software:

ln 10,000 ≈ 9.210

Substitute this into the equation:

L(10) = 1.3 × 9.210

Now, calculate the value:

L(10) ≈ 11.973

So, the estimated runway length needed for an airplane weighing 10,000 pounds is approximately 11.973 thousand feet.

Now, let's address part (b) of the question. The question is asking whether a 20,000-pound airplane needs twice the runway length that a 10,000-pound airplane needs.

To compare the runway lengths, we need to calculate L(x) for both weights.

For a 20,000-pound airplane, we can use the same equation:

L(20) = 1.3 ln 20,000

Using a calculator or software, find the natural logarithm of 20,000:

ln 20,000 ≈ 9.903

Substitute this into the equation:

L(20) = 1.3 × 9.903

Now, calculate the value:

L(20) ≈ 12.874

Therefore, the estimated runway length needed for a 20,000-pound airplane is approximately 12.874 thousand feet.

To determine whether the 20,000-pound airplane requires twice the runway length, we can compare the two values:

12.874 / 11.973 ≈ 1.074

The ratio between the runway lengths is approximately 1.074.

Since the ratio is not exactly 2, we can conclude that a 20,000-pound airplane does not require twice the runway length that a 10,000-pound airplane needs.

Explanation:
The given mathematical model, L(x) = 1.3 ln x, provides a way to estimate runway length based on the weight of the airplane. By substituting the weight into the equation, we can find the estimated runway length. In this case, we used x = 10,000 and x = 20,000 to find the estimated runway lengths for airplanes weighing 10,000 and 20,000 pounds, respectively. Comparing these lengths, we determined that a 20,000-pound airplane does not require twice the runway length of a 10,000-pound airplane.