Jack wants to install a swimming pool that is 12m long and 8m wide. He wants to install a rubberized safety border of uniform width around the pool. The width of the safety border is represented by x and the area of the safety border is 44 m^2.
a) Write an expression in terms of x for the length and width dimensions of the outside safety border.
b) What is the total area of the pool and border?
c) Write a quadratic equation that could be used to calculate the width of the safety border.
d) How wide is the safety border around the pool?
length including border = 12 + 2x
width including border = 8 + 2x
area including border = (12+2x)(8+2x) = .....
area of just the pool = 12(8) = ....
the difference would be the area of the border.
then (12+2x)(8+2x) - 12*8 =44
simplify and solve for x
I assume you know how to solve quadratics, or else you wouldn't be doing this type of question.
Can you find for the width of the border jack can install
a) The length of the outside safety border can be calculated by adding twice the value of x to the original length of the pool. So the expression for the length of the outside safety border is 12m + 2x. Similarly, the width of the outside safety border is 8m + 2x.
b) The total area of the pool and border can be calculated by multiplying the length and width of the outside safety border. So the expression for the total area is (12m + 2x) * (8m + 2x).
c) To write a quadratic equation that can be used to calculate the width of the safety border, we need to set up an equation using the given information. The area of the safety border is given as 44 m^2. We know that the area of a rectangle is given by length multiplied by width. So we can set up the equation (12m + 2x) * (8m + 2x) - 12m * 8m = 44 m^2.
d) To find the width of the safety border, we can solve the quadratic equation. By simplifying and rearranging the equation, we get 4x^2 + 40x - 56 = 0. We can then use the quadratic formula to solve for x. The value of x obtained from the quadratic formula will represent the width of the safety border around the pool.
a) To find the length dimension of the outside safety border, we need to add 2x to the length of the pool. So, the length of the outside safety border is 12 + 2x. To find the width dimension, we need to add 2x to the width of the pool. So, the width of the outside safety border is 8 + 2x.
b) The area of the pool is given by the product of the length and width of the pool, which is 12m * 8m = 96m^2. The area of the safety border is given as 44m^2. To find the total area of the pool and border, we need to add the area of the pool to the area of the safety border. So, the total area is 96m^2 + 44m^2 = 140m^2.
c) Let's denote the width of the safety border as x. The length dimension of the outside safety border is 12 + 2x, and the width dimension is 8 + 2x. The area of the safety border is given as 44m^2. The area of a rectangle is calculated by multiplying its length by its width. So, (12 + 2x)(8 + 2x) = 44. Expanding this equation gives 96 + 24x + 16x + 4x^2 = 44. Simplifying further, we have 4x^2 + 40x + 52 = 0. This is a quadratic equation.
d) To solve the quadratic equation, we can either factor it, complete the square, or use the quadratic formula. In this case, the quadratic equation does not factor easily, so let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)
In our case, a = 4, b = 40, and c = 52. Substituting these values into the quadratic formula, we get:
x = (-40 ± √(40^2 - 4 * 4 * 52))/(2 * 4)
Simplifying further, we have:
x = (-40 ± √(1600 - 832))/8
x = (-40 ± √(768))/8
x = (-40 ± √(6 * 128))/8
x = (-40 ± 8√6)/8
x = -5 ± √6
So, the width of the safety border around the pool can be either -5 + √6 or -5 - √6. Since width cannot be negative in this context, we take the positive value.
Therefore, the width of the safety border around the pool is approximately 0.55 meters.