Question

Imagine that you have an elementary school-age child and are researching childcare options. You find two that you like – one at a child-care center, one is based in a family's home – so it basically comes down to which one will be cheapest. The child-care center charges $300 to enroll, then $79 per week. The family-run childcare charges $180 to enroll, then $93 per week.

a. Setup two functions to represent the cost of each.
b. Which childcare approach will be the cheapest in the short-term, child-care center or family-run?
c. To choose which plan is best, you need some idea of how long your child will be enrolled. Determine the number of weeks at which it becomes more cost-effective to choose the long-term option.

Answer 1

a) Cost of the child-care center: y = 300 + 79x

Cost of the family-run childcare: y = 180 + 93x

Where x is the number of weeks.

b) In the short-term, the family-run childcare will be cheapest because the charge for enrolling is less than for the child-care center.

c)The long term option (child-care center) becomes more cost-effective if you plan to enroll your child for more than 8.6 weeks.

Explanation:

Hi there!

a)Let y be the cost of childcare. In the child-care center, the cost is 300 plus 79 per week. Let x be the number of weeks, then the cost function for the child-care center will be:

y = 300 + 79x

In the same way, the cost of the family-run childcare will be:

y = 180 + 93x

b) In the short-term, the family-run childcare will be cheapest because the charge for enrolling is less than for the child-care center. Let´s calculate the cost of both childcare options for low values of x:

Child-care center:

y = 300 + 79 x

x = 0, y = 300

x = 1, y = 379

x = 2, y = 458

x = 3, y = 537

x = 4, y = 616

Family-run:

y = 180 + 93x

x = 0, y = 180

x = 1, y = 273

x = 2, y = 366

x = 3, y = 459

x = 4, y = 552

c) We have to find the value of x for which the cost of the family-run childcare is greater than the cost of the child-care center:

cost family-run > child-care center

180 + 93x > 300 + 79 x

93x -79x >300 - 180

14x > 120

x > 120/14

x > 8.6

The long term option (child-care center) becomes more cost-effective if you plan to enroll your child for more than 8.6 weeks.