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The Diver Time company makes digital watches for scuba divers. The company models its profits with
the function P(d) = −2d2+13d−6, where d is the number of watches sold, in hundreds, and P(d) is the
company’s profit, in tens of thousands of dollars.
a) How many watches does Diver Time need to sell to maximize profits?
b) What is the maximum profit?
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Part (a)
You are given a quadratic function
P(d) = -2d^2 + 13d - 6. (1)
The function gas a negative leading coefficient -2 at the quadratic term d^2, therefore, the function is a parabola
turned down. It is very well known fact that such a quadratic function gets its maximum at the point
d = ,
where "a" is the coefficient at the squared term and "b" is the coefficient at the linear term.
In your case a= -2, b= 13, therefore the quadratic function gets its maximum value at
= = = 3.25
It means that the company should sell hundreds of watches, or 325 watches.
Part (a) is completed and the answer is 3.25 hundreds of watches should be sold (or 325 watches).
Part (b)
To find the maximum profit, substitute the found value of = 3.25 into the formula (1) to get
= P(3.25) = -2*3.25^2 + 13*3.25 + 6 = 27.125 tens thousands of dollars, or $271,250.
Answer. 325 watches should be sold to get the maximum profit of $271,250.
Solved.
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On finding the maximum/minimum of a quadratic function see the lessons
- HOW TO complete the square to find the minimum/maximum of a quadratic function
- Briefly on finding the minimum/maximum of a quadratic function
- HOW TO complete the square to find the vertex of a parabola
- Briefly on finding the vertex of a parabola
Consider these lessons as your textbook, handbook, tutorials and (free of charge) home teacher.
Learn the subject from there once and for all.
Also, you have this free of charge online textbook in ALGEBRA-I in this site
- ALGEBRA-I - YOUR ONLINE TEXTBOOK.
The referred lessons are the part of this textbook under the topic "Finding minimum/maximum of quadratic functions".
Save the link to this online textbook together with its description
Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson
to your archive and use it when it is needed.