SOLUTION: The revenue achieved by selling x graphing calculators is figured to be x(45-0.5x)dollars. The cost of each calculator is $25. How many graphing calculators must be sold to make

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: The revenue achieved by selling x graphing calculators is figured to be x(45-0.5x)dollars. The cost of each calculator is $25. How many graphing calculators must be sold to make       Log On


   

Question 333541: The revenue achieved by selling x graphing calculators is figured to be
x(45-0.5x)dollars. The cost of each calculator is $25. How many graphing calculators must be sold to make a profit(revenue-cost) of at least 187.50?
Answer by jrfrunner(365) About Me  (Show Source):
You can put this solution on YOUR website!
let r=revenue
let c=cost
let p=profit
let x=number of graphing calculators
---
given
r=x%2845-0.5x%29=45x-0.5%2Ax%5E2
c=25x
p=r-c=45x-0.5%2Ax%5E2-25x
simplify p: p=-0.5%2Ax%5E2%2B20x
---
need to find x, such that p>187.50
==
-0.5%2Ax%5E2%2B20x%3E187.5
-0.5%2Ax%5E2%2B20x-187.5%3E0 (subtract 187.50 from both sides
-x%5E2%2B40x-375%3E0+ (multiply both sides by 2)
--
solve via quadratic solution for x
x=%28-40-sqrt%28%28-40%29%5E2-4%2A%28-1%29%2A%28-375%29%29%29%2F%282%2A%28-1%29%29 or x=%28-40%2Bsqrt%28%28-40%29%5E2-4%2A%28-1%29%2A%28-375%29%29%29%2F%282%2A%28-1%29%29
x=15 or 25 but you need to check for "valid" values
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draw a line
mark the points 15 and 25 on this line
you have created 3 number segments to check for validity of the x values
1. x values from -infinity to 15
2. x values from 15 to 25
3. x values from 25 to + infinity
--
choose a value from each segment, such as 0 for the first segment, 20 for the second segment, and 30 for the third segment.
test these chosen x values against -0.5%2Ax%5E2%2B20x-187.5%3E0
to determine the validity of each segment.
--
you will find that the line segment for x between 15 to 25 that make the profit positive, and that is your answer (between 15 to 25 calculators will yield a proift greater than 187.50)