Math
posted by lola .
An apprentice Carpenter has designed a wooden coffee table. he expects to sell 25 such tables for 95$ each at an upcoming artisan market. by conducting a survey, he determines that for each $2 reduction in the price of the table, he will likely gain three sales.
a) an expression that describes the price of each table, based on the number of reductions, is 95 – 2x , where x represents the number of $2 reductions in price. write a similar expression to describe the number of tables the carpenter can expect to sell, based on the number of reductions.
***I got that part… its 25+3x
b) Write an equation to describe the carpenter’s income as a product of the price of each table and the expected number of tables sold.
**I also found this part. Income= (952x) (25+3x) = 2375+235x 6x2
c)
The carpenter hopes to earn $3600 to pay for his time, materials, and sales booth, as well as make a small profit. Determine the number of $2 reductions he can apply to the price of the tables to make $3600. ?

So you want
(952x) (25+3x) = 3600
2375+235x 6x^2  3600 = 0
6x^2  235x +1225 = 0
by the quadratic formula:
x = 6.19 or x = 32.9
check:
if x = 6, number sold is 25+18 = 43 ,
price of each = 9512 = 83
income = 43(83) = $3569 , close enough to $3600
if x = 33 , number sold = 25 + 124
price of each = 9566 = 29
income = 124(29) = $3596 , same profit
but he would have to work so much harder
so he should sell them at $83 and make 43 of them to get appr $3600
HOWEVER, if you want the max profit,
then
profit = 2375+235x 6x^2  3600
d(profit)/dx = 235  12x = 0 for a max of profit
12x = 235
x = 235/12 = appr 19.58
there should be 20 reductions of $2
Proof:
number sold = 25 + 60 = 85
price of each = 95  40 = 55
income = 85(55) = 4675
take one less, x = 19
number sold = 25 +57 = 82
price of each = 95  38 = 57
income = 82(57) = 4674 , ahh $1 less
take one more, x = 21
number sold = 25 + 63 = 84
price of each = 95  42 = 53
income = 84(53) = 4452 , ahh again less
To make around $3600
there should be 6 reductions of $2
to make a max profit, there should be 20 reductions of $2