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!

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

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.

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

1. To find the price that yields the maximum revenue for the rental business, we need to understand the relationship between the rental price and the number of rentals, as well as the relationship between the number of rentals and revenue.

Let's break down the given information step by step:

- The rental business charges $12 per canoe, and on average, they have 36 rentals per day. This implies that the revenue they generate from canoe rentals per day is 36 * $12 = $432.

- The problem states that for every 50-cent increase in rental price, the business loses two rentals per day. This means that if they increased the rental price by 50 cents, they would have 36 - 2 = 34 rentals. Likewise, for a $1 increase in rental price, they would have 36 - 4 = 32 rentals, and so on.

Now, let's determine the revenue at different rental prices and find the price that yields the maximum revenue:

Rental price: $12, Rentals: 36, Revenue: 36 * $12 = $432
Rental price: $12.50, Rentals: 34, Revenue: 34 * $12.50 = $425
Rental price: $13, Rentals: 32, Revenue: 32 * $13 = $416

By calculating the revenue for different rental prices, we can see that as the rental price increases, the revenue decreases. This is because even though each rental generates more money, the decrease in the number of rentals outweighs the increase in rental price.

Now, let's address the second part of your question regarding "50-2x." The equation "50-2x" represents the number of rentals as a function of the price increase (x). This equation is derived from the given information that for every 50-cent increase in rental price, the business loses two rentals per day. Since the initial number of rentals is 50 (36 rentals per day plus 14 rentals based on the given information), we subtract 2x (2 rentals lost per 50-cent increase in price).

Regarding the vertex, you mentioned that it is (1/2, 600.25), but that is incorrect. The vertex represents the maximum revenue, not the rental price. To find the rental price that yields the maximum revenue, we need a different approach.

Since we already observed that as the rental price increases, the revenue decreases, we want to find the highest rental price that still has revenue greater than $432 (the initial revenue). Let's calculate:

Rental price: $12, Rentals: 36, Revenue: 36 * $12 = $432
Rental price: $12.50, Rentals: 34, Revenue: 34 * $12.50 = $425
Rental price: $13, Rentals: 32, Revenue: 32 * $13 = $416

It seems that $12.50 is the highest rental price for which the revenue is still greater than $432. Therefore, the price that yields the maximum revenue is $12.50, not $10.50 as mentioned in the answer key.

2. In this problem, we are given a 1200-foot roll of fencing and are asked to find the dimensions of the largest rectangular enclosure by splitting it in half.

Let's break it down step by step:

- We have a rectangular enclosure that is split in half, creating two smaller rectangles. Both rectangles have the same dimensions since we are splitting the original enclosure equally.

- To determine the dimensions that yield the largest enclosure, we need to consider the perimeter of the enclosure, which is the sum of all four sides.

- Let's assume the width of the original rectangular enclosure is x. Since we are splitting it in half, each smaller rectangle will have a width of x/2.

- Since the problem states that the total fencing we have available is 1200 feet, the perimeter of the enclosure is given by 2x + 4(x/2) = 1200.

- Simplifying this equation, we get 2x + 2x = 1200, which further simplifies to 4x = 1200.

- By dividing both sides of the equation by 4, we find that x = 300.

- Now we have the width of the original rectangular enclosure, which is 300 feet. Since we split it in half, each smaller rectangle will have a width of 150 feet.

- Using these dimensions, the total fencing used is 2(300) + 2(150) = 900 feet.

Therefore, the dimensions of the largest enclosure are 300 feet by 150 feet, and the total amount of fencing used is 900 feet.