Henry has a patio next to his house that is 15ft by 10ft. He wants to put a uniform flower bed around three sides of the patio. The area of the flower bed is 100 ft^2. What is the width of the flower bed?
Is it 6.67 ft
To solve this problem, we need to find the width of the flower bed.
Let's assume the width of the flower bed is "x" feet.
The dimensions of the original patio are 15ft by 10ft. When we add the flower bed around three sides, the overall dimensions would be increased by twice the width of the bed (since the bed is wrapped around three sides).
So, the dimensions of the new patio (including the flower bed) would be (15+2x)ft by (10+2x)ft.
The area of the new patio (including the flower bed) can be calculated by multiplying the length by the width:
(15+2x) * (10+2x) = 150 + 30x + 20x + 4x^2 = 4x^2 + 50x + 150.
Since the area of the flower bed is given as 100ft^2, we can set up the following equation:
4x^2 + 50x + 150 - 100 = 0.
Simplifying the equation, we get:
4x^2 + 50x + 50 = 0.
To solve this quadratic equation, you can use factoring, completing the square, or the quadratic formula. In this case, factoring is not straightforward, so let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a).
Plugging in the values for a, b, and c into the quadratic formula:
x = (-50 ± √(50^2 - 4 * 4 * 50)) / (2 * 4).
Simplifying further:
x = (-50 ± √(2500 - 800)) / 8
x = (-50 ± √1700) / 8.
To find the values of x, we need to calculate the square root of 1700:
√1700 ≈ 41.23.
Now we can substitute this value back into the equation:
x = (-50 ± 41.23) / 8.
Solving for x gives us two possible values:
x₁ = (-50 + 41.23) / 8 ≈ 0.78.
x₂ = (-50 - 41.23) / 8 ≈ -13.9.
We can ignore the negative value, since the width cannot be negative.
Therefore, the width of the flower bed is approximately 0.78 feet.
let the width of the uniform border be x ft
new width = 10+2x
new length = 15 + 2x
area of whole thing = (10+2x)(15+2x)
(10+2x)(15+2x) - 150 = 100
150 + 50x + 4x^2 - 15 - 100 = 0
4x^2 + 50x - 100 = 0
2x^2 + 25x - 50 = 0
doesn't factor, so use the formula
x = (-25 ± √1025)/4
= 1.754 or some negative
So 1.754 ft must be added to the width and length, so the width of the flower bed must be 1.754 ft
check:
area of patio = 150 ft^2
area of whole thing
= (10+2(1.754)) (15+2(1.754))
= 249.999.. or 250
increase = 250 - 150 = 100
my answer is correct