For a given take-off weight, the take-off distance increases if the air temperature increases. For a specific example, if a KC-135 weighs 200,000 lbs, the take-off distance is modeled by f(t)=1/3(t+145)^2 - 67. If D = f(t) , where D is the take-off distance as a function of t, temperature which is measured in °C and is valid for the temperature domain: [0° to 30°].

Does the inverse exist. If so, interpret the meaning of the inverse function in terms of the relationship between distance and temperature. Within your explanation, include a specific example using correct function notation.


Dont really know where to start on this one. Any help would be greatly appreciated.

solve for t is the first step.

What does the equation mean?

To determine whether the inverse of a function exists, we need to check if the original function is one-to-one, or in other words, if it passes the horizontal line test. If the function does pass the horizontal line test, then the inverse exists.

In this case, the given function is:

D(t) = 1/3(t + 145)^2 - 67

To check if it is one-to-one, we can analyze the graph of the function. However, a simpler way to check is by examining the derivative. If the derivative is always positive or always negative within the given domain, then the function is one-to-one.

Taking the derivative of D(t) with respect to t, we get:

D'(t) = 2/3(t + 145)

Since the derivative is always positive within the given domain of t (0° to 30°), we can conclude that the original function is indeed one-to-one.

Therefore, the inverse of the function exists.

To interpret the meaning of the inverse function in terms of the relationship between distance and temperature, we need to find the inverse function. To do this, we can swap the roles of D and t in the original equation and solve for t.

Swapping D(t) with t and solving for t, we have:

t = 1/3(D + 145)^2 - 67

This equation represents the inverse function. It relates the temperature (t) to the take-off distance (D).

As an example, let's say the take-off distance (D) is 500 units. We can use the inverse function to find the corresponding temperature (t) by substituting D = 500 into the equation:

t = 1/3(500 + 145)^2 - 67

Evaluating this expression will give us the temperature value (t) associated with a take-off distance of 500 units.