Quadratic word problem (sales for max profit)

[Saki]

A manufacturer has been selling light bulbs for $2.00 each. People have been buying 100 light bulbs per month. The manufacturer belives that for every $1.00 increase per bulb, 4 fewer bulbs will be sold each month. It costs the manufacturer $.50 to make each light bulb. Assume the manufacturer sells every light bulb that he makes and that any whole number of light bulbs can be made.

1 } What price should the manufacturer sell each light bulb to make the maximum profit each month?

2 } How many light bulbs will maximize the manufacturer's monthly profit?

3 } What is the manufacturer's monthly profit?

My teacher said to use parabola's for illustrating this, can someone please start me off on this problem?

 

[tkhunny]

Re: Quadratic word problem

A manufacturer has been selling light bulbs for $2.00 each. People have been buying 100 light bulbs per month. The manufacturer belives that for every $1.00 increase per bulb, 4 fewer bulbs will be sold each month. It costs the manufacturer $.50 to make each light bulb. Assume the manufacturer sells every light bulb that he makes and that any whole number of light bulbs can be made.

1 } What price should the manufacturer sell each light bulb to make the maximum profit each month?

2 } How many light bulbs will maximize the manufacturer's monthly profit?

3 } What is the manufacturer's monthly profit?

My teacher said to use parabola's for illustrating this, can someone please start me off on this problem?

It's not really the greatest advice to say, "Use a parabola." You should model the situation using all your algebra skills and see what pops out. If it happens to be a parabola-type object, great, because we know all about those, right?

This one is pretty much common sense. Don't over think it because you don't know a magic formula.

x = Price Change
Price of bulb = 2.00 + x
Bulbs Sold = 100 - 4*x
Income = (Price of Bulb) * (Bulbs Sold) = [(2.00 + x) * (100 - 4*x)]
Cost to Manufacture = (100-4*x)*0.50
Profit = [Income - (Cost to Manufacture)] = ((2.00 + x) * (100 - 4*x)) - ((100-4*x)*0.50)

Do some simplifying. See what stuff pops out.

Note: This is a little bit tricky. The problem statement suggests price changes of $1.00. Does this mean you can change by parts of $1.00? It may, but it may not. If you cannot use partial price changes, the maximum or minimum value may be a bit trickier to find. If it were me, I'd state up front, "Assume price changes can be of any magnitude."

 

[Saki]

Re: Quadratic word problem

It's not really the greatest advice to say, "Use a parabola." You should model the situation using all your algebra skills and see what pops out. If it happens to be a parabola-type object, great, because we know all about those, right?

This one is pretty much common sense. Don't over think it because you don't know a magic formula.

x = Price Change
Price of bulb = 2.00 + x
Bulbs Sold = 100 - 4*x
Income = (Price of Bulb) * (Bulbs Sold) = [(2.00 + x) * (100 - 4*x)]
Cost to Manufacture = (100-4*x)*0.50
Profit = [Income - (Cost to Manufacture)] = ((2.00 + x) * (100 - 4*x)) - ((100-4*x)*0.50)

Do some simplifying. See what stuff pops out.

Note: This is a little bit tricky. The problem statement suggests price changes of $1.00. Does this mean you can change by parts of $1.00? It may, but it may not. If you cannot use partial price changes, the maximum or minimum value may be a bit trickier to find. If it were me, I'd state up front, "Assume price changes can be of any magnitude."

WOW, thank you so much, i just got everything you just said! I sort of understand it now, ill see if i can solve the problem.

 

[Gene]

Or:
P is the price he sells them for.
Q is how many he sells.
He will sell Q = 100 - 4(P-2)
He will net N=(P-.50)*Q
Make the quadratic from
N=(P-.5)(100 - 4(P-2))
Should give the same answer.