1. The area of a rectangular garden in square feet is x^2-9x-400. What is the width and length of the garden?
*I factored this one and got (x+16)(x-25), but I don't know which one is the width and which one is the length.
2. A rectangular pool is 15 ft wide and 40 ft long. The pool is surrounded by a walkway. the walkway is the same width all the way around the pool. The total area of the walkway is 456 square feet. How wide is the walkway?
*I'm not sure how to set up the equation, but is it 456=(15+2x)(40+2x)-(15)(40)?
*I'm not sure how to solve these. Can you please help me?
The problem asks you to find the value of k that wuold make x^2-kx+64 a perfect square trinomial. You said k=12. Your friend said k=6. Who is correct? What mistake was made by the incorrect person?
You work at a restaurant whose weekly profit is given by the formula p=-c^2+14c+800, where c is the average price of the food, in dollars. The manager wants to add delivery service, which will cost the restaruant d=5c+300 per week.
Find the highest average price d the restaurant can sell its food at and still make a profit if they add delivery.
What will the weekly profit p be if the restaurant sells its food at this average price and doesn't offer delivery
#1, Usually, we consider the length to be the larger number.
so comparing x+16 and x-25, we have 3 cases:
a) they are equal: x+16 = x-25 --> 16 = -25, which is false
b) x+16 < x-25 --> 16 < -25 , which is also false
c) x+16 > x-25, ---> 16 > -25, which is true, so
x+16 is the length, and x-25 is the width
#2, your equation is correct, all you have to do is solve for x
#3 , (x-a)^2 = x^2 - 2ax + a^2
so for x^2-kx+64 , a^2 = 64, then a = 8 and
-kx = -2ax
kx = 2(8)x
k = 16
#4
profit = -c^2+14c+800 + 5c+300
= -c^2 + 19c + 1100
= -(c^2 - 19c - 1100)
= -(c - 44)(c + 25)
break-even is when -(c - 44)(c + 25) = 0
or (c - 44)(c + 25) = 0
c = 44 or x = -25 , but c is price of food, thus c > 0
Does that help in reaching a conclusion?
Try a value above 44 and a value of c below 44 and find the profit using the factored equation for profit.
I am sure you can now find how to do the last problem.
1. To find the width and length of the garden given the quadratic equation x^2-9x-400, we need to write it in factored form. You correctly factored the equation as (x+16)(x-25).
In this case, (x+16) represents the width, and (x-25) represents the length.
So, the width of the garden is x + 16, and the length is x - 25.
2. To find the width of the walkway, we can set up the equation using the given information.
Let's assume the width of the walkway is x.
The dimensions of the pool, including the walkway, can be expressed as: width = 15 + 2x and length = 40 + 2x.
The area of the pool including the walkway is given by (15 + 2x)(40 + 2x).
Since the total area of the walkway is 456 square feet, we can set up the equation:
(15 + 2x)(40 + 2x) - (15)(40) = 456
Simplifying the equation, we have:
(15 + 2x)(40 + 2x) - 600 = 456
Expanding and rearranging:
600 + 70x + 4x^2 - 600 = 456
4x^2 + 70x - 456 = 0
Now, you can solve this quadratic equation to find the width of the walkway, x.
1. The area of the rectangular garden can be factored as (x+16)(x-25). To determine the width and length of the garden, we need to understand that the factors represent the dimensions of the garden. In this case, (x+16) represents the width and (x-25) represents the length.
2. To determine the width of the walkway, we can set up the equation based on the information given. Let's assume the width of the walkway is 'x'. Since the walkway surrounds the pool, the width of the garden including the walkway will be the sum of the pool width and twice the walkway width (2x), and the length will be the sum of the pool length and twice the walkway width (2x) as well.
The total area of the garden including the walkway is equal to the area of the entire rectangular space minus the area of just the pool. We can express this mathematically as follows:
Total area = (15 + 2x)(40 + 2x) - (15)(40)
By given information, the total area of the walkway is 456 square feet, so we can set up the equation:
456 = (15 + 2x)(40 + 2x) - (15)(40)
Simplifying this equation further will provide the solution to the width of the walkway.