Question

Omega just got her pilot's license and wants to rent a plane. The Platinum Plane Company charges $180 plus $92 per hour to rent a plane. The Plastic Plane Company charges $250 plus $78 per hour.

a. For what number of hours would the companies charge the same amount?

b. What would the charge be for that number of hours?

Answer 1

Let \( h \) be the number of hours Omega rents the plane.

For the Platinum Plane Company, the total cost can be represented by the equation:
\[
P(h) = 180 + 92h
\]

For the Plastic Plane Company, the total cost can be represented by the equation:
\[
C(h) = 250 + 78h
\]

To find the number of hours \( h \) at which the costs are the same, we set the two equations equal to each other:
\[
180 + 92h = 250 + 78h
\]

Now, we can solve for \( h \):

1. Subtract \( 78h \) from both sides:
\[
180 + 92h - 78h = 250
\]
\[
180 + 14h = 250
\]

2. Subtract 180 from both sides:
\[
14h = 250 - 180
\]
\[
14h = 70
\]

3. Divide both sides by 14:
\[
h = \frac{70}{14} = 5
\]

Thus, the companies charge the same amount when \( h = 5 \) hours.

Next, we can calculate the charge for that number of hours:

Using either equation (let's use the Platinum Plane Company's equation):
\[
P(5) = 180 + 92 \cdot 5 = 180 + 460 = 640
\]

or using the Plastic Plane Company's equation:
\[
C(5) = 250 + 78 \cdot 5 = 250 + 390 = 640
\]

Both calculations confirm the cost.

**Final Answers:**
a. The number of hours where the companies charge the same amount is \( 5 \) hours.

b. The charge for that number of hours is \( \$640 \).