Brittany is checking whether the pool she wants to buy will fit in her backyard. The pool with the patio measures 4 m by 5 m. Refer to the diagram below.
a) Write an algebraic expression, in factored form, for the shaded area.
b) When Brittany measures her yard, she notices that x = 3, but she needs 20 m2 for her dog to run. Will the pool fit in her yard?
a) To find the shaded area, we need to subtract the area of the patio from the total area of the pool. The total area of the pool is given by its dimensions of 4 m x 5 m, which equals 20 m².
The area of the patio is not given, so we need to express it in terms of x. Let's assume x represents the width of the patio. Since the length of the patio is 5 m (the same as the length of the pool), the width of the patio would be (5 - x) m.
The area of the patio is then (5 - x) * x = 5x - x².
Subtracting the area of the patio from the total area of the pool gives us the shaded area: 20 - (5x - x²).
b) To check if the pool will fit in her yard, we need to find the value of x that satisfies the expression for the shaded area to be at least 20 m².
So, we have the inequality: 20 - (5x - x²) ≥ 20
Simplifying the inequality, we get: -5x + x² ≥ 0
To solve this inequality, we can factor it: (x - 0)(x - 5) ≥ 0
Now we have two critical points: x = 0 and x = 5. We need to test the intervals between these points to determine when the inequality is true.
Testing the interval (0, 5), we choose a value in the interval, such as x = 3. Plugging it into the inequality, we get: (3 - 0)(3 - 5) ≥ 0.
Simplifying this expression gives: 3 * (-2) ≥ 0, which is -6 ≥ 0. Since -6 is not greater than or equal to zero, this interval does not satisfy the inequality.
Therefore, we conclude that the pool will not fit in Brittany's yard, as there is no value of x that will make the shaded area greater than or equal to 20 m².