SOLUTION: 18) The We Make Widgets Company manufactures widgets. They find that when they charge d dollars for each widget, their income, I(d) can be expressed by the formular I(d) = -120d^2

Question 173539This question is from textbook Amsco's Preparing for the Regents Examination Mathematics B
: 18) The We Make Widgets Company manufactures widgets. They find that when they charge d dollars for each widget, their income, I(d) can be expressed by the formular I(d) = -120d^2 + 14,400d + 100. What price should they charge to maximize their income? If this price is charged, what is their maxium income?
Please can someone help me and show work, step by step?
Thank you so much for your time.
Happy Holidays This question is from textbook Amsco's Preparing for the Regents Examination Mathematics B

Found 2 solutions by nerdybill, josmiceli:
Answer by nerdybill(7384) (Show Source): You can put this solution on YOUR website!
18) The We Make Widgets Company manufactures widgets. They find that when they charge d dollars for each widget, their income, I(d) can be expressed by the formular I(d) = -120d^2 + 14,400d + 100. What price should they charge to maximize their income? If this price is chargeThe d coordinate = -14400/2(-120)
d, what is their maxium income?
.
Looking at the 'a' coefficient of the given equation:
I(d) = -120d^2 + 14400d + 100
.
We see that it is negative -- meaning that it is a parabola that is opened downward. So, if we simply find the "vertex" of the equation, we'll find the "maximum".
.
For additional info on vertex form of a parabola:
http://www.mathwarehouse.com/geometry/parabola/standard-and-vertex-form.php
.
The vertex is located at:
The d coordinate = -b/2a
The d coordinate = -14400/2(-120)
The d coordinate = -14400/(-240)
The d coordinate = 60
What price should they charge to maximize their income? $60
.
what is their maxium income?
Plug it back into the original formula:
.
I(d) = -120d^2 + 14,400d + 100
I(60) = -120(60)^2 + 14,400(60) + 100
I(60) = -120(3600) + 14400(60) + 100
I(60) = -432000 + 864000 + 100
I(60) = 432000 + 100
I(60) = $432,100

Answer by josmiceli(19441) (Show Source): You can put this solution on YOUR website!
I always think of the maximum as being exactly between
the 2 roots of a parabola (where it crosses the x-axis
The quadratic formula finds the roots

There is a (+) and a (-) answer, and thoses are the roots
If you rewrite the formula like this:

you can see that is in the middle and you add
the 2nd term to get the larger root and subtract the 2nd
term to get the smaller root.
So, you just have to find to find the maximum
The general formula for finding roots is

The equation in the problem is ( replaces )

This says when is a maximum
$60 /widget should be charged to maximize income
----------------
The problem wants to know what that income is

The maximum income is $432,100
You can check the answers by making a little
bit less, say and finding
and a little bit more, say and finding
In both cases, should be
less than
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