Question

Not sure where to start with this question :
A sports store sells 70 pairs of shoes per day for $40 each. The owner estimates that for each $3 increase in price, 3 fewer sales are made each day. What price should be charged to maximize profit?

Answer 1

@phi

Answer 2

try to write down equations

Answer 3

How? like 70 pairs a day = 70x? and then the cost of 40? like 70x=40? This is my first assignment on this so im just trying to figure it out

Answer 4

That is the hard part...

I would focus on
$3 increase in price, 3 fewer sales
which is equivalent to $1 up in price causes 1 fewer sale
if we call the price increase x
then we can say
total = (40+x)(70-x)
for x=0, we get the original conditions.

we have to find the "x" value so that total is as big as possible

Answer 5

Okay, with you so far

Answer 6

so solve for x?

Answer 7

once you have an equation, it's math.

there are two ways: use calculus to find a min/max
or in this case, the equation is a parabola, with a \( \cap \) shape, so the max value is its vertex

Answer 8

Okay ill put the equation into my caculator

Answer 9

I think I did something wrong

Answer 10

i put it in and got x=15 +- the square root of 3025-t @phi

Answer 11

or 55-t simplified

Answer 12

what is t ?

Answer 13

the total

Answer 14

would the vertex be 15?

Answer 15

if you use y instead of t (so it looks "normal")
y= -x^2 +30x +2800

Answer 16

for x

Answer 17

oh okay

Answer 18

and the vertex for that equation would be -15 right?

Answer 19

yes, the vertex is at -b/(2a)
where a, b (and c) are the coefficients:
y = a x^2 + b x + c
matching that with
y= -x^2 +30x +2800

a is -1 (-x^2 means -1 * x^2)
b is 30
and -b/(2a) is -30/(2*-1) = 15

Answer 20

so x=15 (not -15)

Answer 21

oh okay I see i second guessed myself

Answer 22

thank you so much for your help! its a little clearer now.

Answer 23

you still have to answer the question:
What price should be charged to maximize profit?

Answer 24

to find that out you'd substitute in 15 for x in the equation, right?

Answer 25

that would give the total dollars.
they want the cost of one pair

Answer 26

if you look at the "start" equation
total = (40+x)(70-x)

which of those represents the cost of one pair ?

Answer 27

the 40+ x?

Answer 28

yes

Answer 29

so the total would be 55?

Answer 30

no wait

Answer 31

sorry that doesnt make sense

Answer 32

the price is 55
(and the number of pairs sold will be 70 - 15 = 55)
and at the total profit would be 55*55 = 3025

Answer 33

oh really? okay haha i guess i should stop second guessing myself sorry

Answer 34

yes 3025 is what i got too

Answer 35

so the answer would be 55 then

Answer 36

yes,
What price should be charged to maximize profit? $55 per pair.

Answer 37

Okay, gotcha. Thank you so much!