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29 December, 23:36

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.

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  1. 30 December, 02:58
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    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.

    Step-by-step 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.
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