A landscaper is designing a flower garden in the shape of a trapezoid. She wants the shorter base to be 3 yards greater than the height and the longer base to be 7 yards greater than the height. She wants the area to be 295 square yards. The situation is modeled by the equation h2 + 5h = 295. Use the quadratic formula to find the height that will give the desired area. Round to the nearest hundredth of a yard.
a 17.36 yards
b 600 yards
c 14.86 yards
d 29.71 yards
To solve the quadratic equation h^2 + 5h = 295, we can rearrange it to h^2 + 5h - 295 = 0.
Using the quadratic formula, which is given by x = (-b ± √(b^2 - 4ac)) / (2a), for the quadratic equation ax^2 + bx + c = 0, we can plug in the values a = 1, b = 5, and c = -295:
h = (-5 ± √(5^2 - 4(1)(-295))) / (2(1))
Simplifying this expression gives:
h = (-5 ± √(25 + 1180)) / 2
h = (-5 ± √1205) / 2
Now we can calculate the two possible values for h:
h1 = (-5 + √1205) / 2 ≈ 17.36 yards
h2 = (-5 - √1205) / 2 ≈ -22.36 yards
Since height cannot be negative in this context, we discard the second solution and round the first solution to the nearest hundredth of a yard.
Therefore, the height that will give the desired area is approximately 17.36 yards.
The correct option is a. 17.36 yards.
To find the height that will give the desired area of 295 square yards, we need to solve the quadratic equation h^2 + 5h = 295 using the quadratic formula.
The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, our equation is h^2 + 5h = 295, where a = 1, b = 5, and c = -295. Plugging these values into the quadratic formula, we get:
h = (-5 ± √(5^2 - 4(1)(-295))) / (2(1))
Simplifying further:
h = (-5 ± √(25 + 1180)) / 2
= (-5 ± √(1205)) / 2
Now, we can calculate the two possible values for h:
h = (-5 + √(1205)) / 2 ≈ 14.86 yards
h = (-5 - √(1205)) / 2 ≈ -19.86 yards
Since the height of a flower garden cannot be negative, we round the positive value to the nearest hundredth:
The height that will give the desired area is approximately 14.86 yards.
Therefore, the correct answer is option c) 14.86 yards.
To find the height that will give the desired area, we can use the quadratic formula.
The equation that models the situation is: h^2 + 5h = 295
To use the quadratic formula, we need to first rearrange the equation to the form "ax^2 + bx + c = 0." In this case, the equation is already in that form.
Now we can identify the values of a, b, and c in the equation:
a = 1
b = 5
c = -295
The quadratic formula is: x = (-b ± √(b^2 - 4ac)) / (2a)
Substituting the values into the formula, we get:
h = (-5 ± √(5^2 - 4(1)(-295))) / (2(1))
Simplifying further:
h = (-5 ± √(25 + 1180)) / 2
h = (-5 ± √1205) / 2
Now, let's calculate the approximate values of h using the quadratic formula:
h ≈ (-5 + √1205) / 2 ≈ 14.86
h ≈ (-5 - √1205) / 2 ≈ -19.86
Since we are looking for the height, we discard the negative value.
Therefore, the height that will give the desired area is approximately 14.86 yards.
The correct answer is option c) 14.86 yards.