Pre-calc

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Please explain. I want to understand the steps.

1. A frog jumps from a rock to the shore of a pond. Its path is given by the equation y=(-5/72)x^2+(5/3)x, where x is the horizontal distance in inches, and y is the height in inches. What is the frog's maximum height? How far had it traveled horizontally when it reached its maximum height?

2. A local bakery determines that when the price of a package of cookies is $12, it sells on average 220 packages of cookies per day. For every $0.50 increase in the price, it sells on average 10 packages of cookies less. Determine the price that maximizes the bakery's profit. Determine the maximum profit.

3.Find the quadratic function with the vertex (7,10) and with point (10,28) on its graph.

  • Pre-calc -

    looks like you are studying quadratic function.
    You should have learned a method to find the vertex of such a funtion
    here is a quick way
    for y = ax^2 + bx + c
    the x of the vertex is -b/(2a)
    so in your case
    x = (-5/3)/(-5/32) = 32/3
    sub back in to get
    y = 80/9

    y is your height,
    x is your horizontal distance

    2. let the number of $0.50 increases be n

    profit = (12 + .5n)(220 - 10n)

    expand to get a quadratic, repeat the method I showed you in #1

    3. let the function be
    y = a(x-7)^2 + 10 , in the usual vertex form
    but (10,28) lies on it
    so 28 = a(10-7)^2 + 10

    solve for a, and you got it

  • Pre-calc -

    For 2 I did the function and got n=-1. then i substitute it in and got 11.5 is that the answer? if so to which of the 2 questions?

  • Pre-calc -

    You are right, n = -1

    Since I defined n as the number of .50 "increases" we have to interpret that as a "decrease" of 50 cents.

    you can verify this by making a chart

    cost - number - profit
    12.00 220 ---- 2640
    12.50 210 ---- 2625
    11.50 230 ---- 2645
    11.00 240 ---- 2640
    etc

    so the price should be $11.50, selling 230 of them for a maximum profit of $2645

    btw, (11.5 , 2645) would be the vertex of your function

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