Question

What is the minimum number of hours Sherry needs to rent a plane from either company to make the total cost equal?

Answer 1

To find the minimum number of hours Sherry needs to rent a plane from either company so that the total costs are equal, we first need to define the cost structure for each company.

Let's say:
- Company A charges a flat fee of \( c_A \) and an hourly rate of \( r_A \) per hour.
- Company B charges a flat fee of \( c_B \) and an hourly rate of \( r_B \) per hour.

The total cost for renting from Company A for \( h \) hours can be expressed as:
\[
\text{Cost}_A = c_A + r_A \cdot h
\]

Similarly, the total cost for renting from Company B for \( h \) hours is:
\[
\text{Cost}_B = c_B + r_B \cdot h
\]

To find when these costs are equal, we set them equal to each other:
\[
c_A + r_A \cdot h = c_B + r_B \cdot h
\]

Rearranging this equation, we get:
\[
c_A - c_B = (r_B - r_A) \cdot h
\]

From this, we can solve for \( h \):
\[
h = \frac{c_A - c_B}{r_B - r_A}
\]

This equation gives us the number of hours \( h \) at which the costs are equal. To ensure that the value of \( h \) is sensible (i.e., a non-negative number), we should also consider the signs of \( c_A - c_B \) and \( r_B - r_A \):
- If \( r_B - r_A = 0 \), then \( h \) would be undefined unless \( c_A - c_B = 0 \) as well (in which case costs are always equal).
- If \( c_A - c_B \) and \( r_B - r_A \) have opposite signs, it would imply that one company's costs increase faster than the other's due to their rates.

Therefore, the minimum number of hours \( h \) that Sherry needs to rent from either company is given by:
\[
h = \frac{c_A - c_B}{r_B - r_A}
\]
provided \( r_B \neq r_A \) and the signs of the numerator and denominator allow for a non-negative value of \( h \).

Make sure to provide specific values for \( c_A \), \( c_B \), \( r_A \), and \( r_B \) if you want a numerical answer!