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41. FINANCIAL LITERACY A babysitting club sits for 50 different families. They would like to
increase their current rate of $9.50 per hour. After surveying the families, the club finds
that the number of families will decrease by about 2 for each $0.50 increase in the
hourly rate.
a. Write a quadratic function that models this situation.
b. State the domain and range of this function as it applies to the situation.
c. What hourly rate will maximize the club's income? Is this reasonable?
d. What is the maximum income the club can expect to make?
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Respuesta :

Answer:

maximum at 7.50 rate with 484

Step-by-step explanation:

Given that a babysitting club sits for 50 different families. They would like to

increase their current rate of $9.50 per hour.

As of now revenue =[tex]50(9.50) = 475[/tex]

Also given that after surveying the families, the club finds  that the number of families will decrease by about 2 for each $0.50 increase in the  hourly rate.

Let x be the no of times 0.50 increase is affected.  Then no of families would be 50-2x

Hence revenue as a function of x is

f(x) =[tex](9.50+0.50x)(50-2x)\\= 475 +6x-x^2[/tex]

x can take values between 0 and 25.

Range can be (0,484)

(vertex of parabola of f(x) since open down)

c) [tex]f'(x) = 6-2x\\f"(x) =-2<0[/tex]

Equate I derivative to 0

x =3

Hourly rate at [tex]9-3(0.50)\\=7.50[/tex]

will maximize the revenue.

Yes. reasonable

because within the domain

d) Maximum income is 484

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