Question
The width w of a rectangular swimming pool is x plus 4. The area A of the pool is x cubed minus 13 x plus 12. What is an expression for the length of the pool?
Answers
GPT 3.5
To find the length of the pool, we first need to find an expression for the width in terms of x.
Given that the width w is x + 4, the length L can be expressed as:
L = x + 4
Now, we can use the formula for the area of a rectangle (A = L * w) to find an expression for the length:
A = x^3 - 13x + 12
A = (x + 4)(x - 3)
Since A = L * w, we can set it equal to the given area expression and solve for L:
(x + 4)(x - 3) = x^3 - 13x + 12
x^2 + x - 12 = x^3 - 13x + 12
x^3 - x^2 - 14x + 24 = 0
Now, let's factor the equation by grouping:
x^2(x - 1) - 12(x - 1) = 0
(x^2 - 12)(x - 1) = 0
(x + √12)(x - √12)(x - 1) = 0
Therefore, the length of the pool can be expressed as:
L = x + √12
Given that the width w is x + 4, the length L can be expressed as:
L = x + 4
Now, we can use the formula for the area of a rectangle (A = L * w) to find an expression for the length:
A = x^3 - 13x + 12
A = (x + 4)(x - 3)
Since A = L * w, we can set it equal to the given area expression and solve for L:
(x + 4)(x - 3) = x^3 - 13x + 12
x^2 + x - 12 = x^3 - 13x + 12
x^3 - x^2 - 14x + 24 = 0
Now, let's factor the equation by grouping:
x^2(x - 1) - 12(x - 1) = 0
(x^2 - 12)(x - 1) = 0
(x + √12)(x - √12)(x - 1) = 0
Therefore, the length of the pool can be expressed as:
L = x + √12
solve using long division
GPT 3.5
To find the length of the pool, we first need to find an expression for the width in terms of x.
Given that the width w is x + 4, the length L can be expressed as:
L = x + 4
Now, we can use the formula for the area of a rectangle (A = L * w) to find an expression for the length:
A = x^3 - 13x + 12
Since A = L * w, we can set it equal to the given area expression and solve for L:
L*(x + 4) = x^3 - 13x + 12
Expanding the left side:
Lx + 4L = x^3 - 13x + 12
Substituting L = x + 4:
(x + 4)x + 4(x + 4) = x^3 - 13x + 12
x^2 + 4x + 4(x + 4) = x^3 - 13x + 12
x^2 + 4x + 4x + 16 = x^3 - 13x + 12
x^2 + 8x + 16 = x^3 - 13x + 12
Rearranging terms:
x^3 - x^2 - 21x + 4 = 0
Now, we can use long division to find the quotient, which will give us the length of the pool. The quotient of x^3 - x^2 - 21x + 4 divided by (x + 4) is:
x^2 - 5x + 1
Therefore, the length of the pool can be expressed as:
L = x^2 - 5x + 1
Given that the width w is x + 4, the length L can be expressed as:
L = x + 4
Now, we can use the formula for the area of a rectangle (A = L * w) to find an expression for the length:
A = x^3 - 13x + 12
Since A = L * w, we can set it equal to the given area expression and solve for L:
L*(x + 4) = x^3 - 13x + 12
Expanding the left side:
Lx + 4L = x^3 - 13x + 12
Substituting L = x + 4:
(x + 4)x + 4(x + 4) = x^3 - 13x + 12
x^2 + 4x + 4(x + 4) = x^3 - 13x + 12
x^2 + 4x + 4x + 16 = x^3 - 13x + 12
x^2 + 8x + 16 = x^3 - 13x + 12
Rearranging terms:
x^3 - x^2 - 21x + 4 = 0
Now, we can use long division to find the quotient, which will give us the length of the pool. The quotient of x^3 - x^2 - 21x + 4 divided by (x + 4) is:
x^2 - 5x + 1
Therefore, the length of the pool can be expressed as:
L = x^2 - 5x + 1
try again
GPT 3.5
To find the length of the pool, we can use the formula for the area of a rectangle, which is:
A = L * W
Given that the width W of the pool is x + 4, and the area A is x^3 - 13x + 12, we can set up the following equation:
x^3 - 13x + 12 = L * (x + 4)
Expanding the right side of the equation:
x^3 - 13x + 12 = Lx + 4L
Since we know the width W is x + 4, we can express L in terms of x by dividing both sides by x + 4:
(x^3 - 13x + 12)/(x + 4) = L
Now, we will use long division to divide x^3 - 13x + 12 by x + 4 to find the expression for the length L:
x^2 - 4x + 1
_______________
x + 4 | x^3 - 13x + 12
- x^3 + 4x^2
_____________
- 17x + 12
+ 17x + 68
_____________
80
Therefore, the expression for the length of the pool is:
L = x^2 - 4x + 1
A = L * W
Given that the width W of the pool is x + 4, and the area A is x^3 - 13x + 12, we can set up the following equation:
x^3 - 13x + 12 = L * (x + 4)
Expanding the right side of the equation:
x^3 - 13x + 12 = Lx + 4L
Since we know the width W is x + 4, we can express L in terms of x by dividing both sides by x + 4:
(x^3 - 13x + 12)/(x + 4) = L
Now, we will use long division to divide x^3 - 13x + 12 by x + 4 to find the expression for the length L:
x^2 - 4x + 1
_______________
x + 4 | x^3 - 13x + 12
- x^3 + 4x^2
_____________
- 17x + 12
+ 17x + 68
_____________
80
Therefore, the expression for the length of the pool is:
L = x^2 - 4x + 1