1.R(x) = (12 -0.5x)(36 - 2x)
= 432 – 6x – x^2
= -x^2 -6x + 432
h = -b/2a
= 6/-2 = -3
= -(3)^2 -6(3) +432
= -9 + 18 + 432
= 441
(-3, 441)
12 -1.50 = 10.5
2. 2x + 3y = 1200
= x + 1.5y = 600
x = - 1.5y + 600
A = LW
= (-1.5y + 600)y
= -1.5y^2 + 600y
h = -b/2a
= -600/2(-1.5) = 200
x = -1.5(200) + 600
x = 300
1. A rental business charges $12 per canoe and averages 36 rentals a day. For every 50-cent increase in rental price, it will lose two rentals a day. What price would yield the maximun revenue? I was told the answer by Reiny, but I couldn't get the right answer. I found the vertex- (1/2, 600.25) but my answer key says $10.50. How? Also, why is the new # rented 50-2x? Wouldn't it be 36-2x or something like that?
2. You have a 1200-foot roll of fencing and a large field. You want to make two paddocks by splitting a rectangular enclosure in half. What are the dimensions of the largest such enclosure? I got the right answer, but I just wanted to know how you got the equation part- what does the 2y and 3x represent?
Thanks again Reiny!
4 answers
ok cool! Thanks! But I have a question, why do we have to find the 'h'(-b/2a)? What does the vertex represent in these cases?
Yes Wendy, I made the error in copying it from paper to here , should have been 36-2x
Look at Kudums solution, he did it correctly using the same variable setup as mine
As for #2 (#3 in your earlier post), compare Kudum's with mine and see that we just interchanged our definitions.
I will use my earlier definitions so you can compare how the two different solutions compare:
Make a sketch of a rectangel and draw in a third width to cut in in half as requested in the question.
I let the width be x, and there are 3 of them
I let the entire length be y, there are 2 of those
So 3x + 2y = 1200
This equation relates the x and y, I solved for y
2y = 1200-3x , divide by 2
y = 600 - 1.5x , I used 600 - (3/2)x in my previous post, same thing
Area = xy
= x(600-1.5x)
= 600x - 1.5x^2
the x of the vertex is -b/(2a) = -600/-3 = 200
when x = 200
y = 600 - 1.5(200) = 300
So the rectangle is 300 ft long, and 200 ft wide
Notice Kudum's x and y values are reversed from mine, it all depends how we originally define the variables.
Look at Kudums solution, he did it correctly using the same variable setup as mine
As for #2 (#3 in your earlier post), compare Kudum's with mine and see that we just interchanged our definitions.
I will use my earlier definitions so you can compare how the two different solutions compare:
Make a sketch of a rectangel and draw in a third width to cut in in half as requested in the question.
I let the width be x, and there are 3 of them
I let the entire length be y, there are 2 of those
So 3x + 2y = 1200
This equation relates the x and y, I solved for y
2y = 1200-3x , divide by 2
y = 600 - 1.5x , I used 600 - (3/2)x in my previous post, same thing
Area = xy
= x(600-1.5x)
= 600x - 1.5x^2
the x of the vertex is -b/(2a) = -600/-3 = 200
when x = 200
y = 600 - 1.5(200) = 300
So the rectangle is 300 ft long, and 200 ft wide
Notice Kudum's x and y values are reversed from mine, it all depends how we originally define the variables.
in any parabola written in the form
y = a(x-p)^2 + q, the vertex is (p,q)
if a is positive, q will be a minimum when x = p
if a is negative, q will be a maximum when x = p
if the equation is written in the form
y = ax^2 + bx + c,
the x of the vertex can be simply found by -b/(2a)
once you have that , just sub it back into the equation to find the y of the vertex.
e.g.
y = 3(x-5)^2 + 10 has a vertex of (5,10)
the minimum value of y is 10, when x = 5
If I expand this I get
y = 3x^2 -30x + 85 , but my vertex is no longer obvious by just looking at it
but I can find the x of the vertex by -b/(2a)
-(-30)/(2x3) = 30/6 = 5 , same as before
y = a(x-p)^2 + q, the vertex is (p,q)
if a is positive, q will be a minimum when x = p
if a is negative, q will be a maximum when x = p
if the equation is written in the form
y = ax^2 + bx + c,
the x of the vertex can be simply found by -b/(2a)
once you have that , just sub it back into the equation to find the y of the vertex.
e.g.
y = 3(x-5)^2 + 10 has a vertex of (5,10)
the minimum value of y is 10, when x = 5
If I expand this I get
y = 3x^2 -30x + 85 , but my vertex is no longer obvious by just looking at it
but I can find the x of the vertex by -b/(2a)
-(-30)/(2x3) = 30/6 = 5 , same as before
4 answers