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.
(1 point)
Responses
17.36 yards
17.36 yards
600 yards
600 yards
14.86 yards
14.86 yards
29.71 yards
To find the height that will give the desired area, we can use the quadratic formula:
h^2 + 5h = 295
Rearranging the equation to standard quadratic form, we get:
h^2 + 5h - 295 = 0
Plugging the values into the quadratic formula, we have:
h = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 1, b = 5, and c = -295. Plugging in these values, we get:
h = (-5 ± √(5^2 - 4(1)(-295))) / 2(1)
Simplifying further, we have:
h = (-5 ± √(25 + 1180)) / 2
h = (-5 ± √(1205)) / 2
To find the approximate value of h, we can use a calculator. Rounding to the nearest hundredth, we have:
h ≈ 17.36 yards
Therefore, the correct answer is:
17.36 yards
To solve the quadratic equation, we can use the quadratic formula which is given as:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, the equation we are trying to solve is:
h^2 + 5h = 295
To put it in standard form, we can subtract 295 from both sides:
h^2 + 5h - 295 = 0
Comparing this equation with ax^2 + bx + c = 0, we find that a = 1, b = 5, and c = -295.
Now we can substitute these values into the quadratic formula:
h = (-5 ± √(5^2 - 4 * 1 * -295)) / (2 * 1)
Simplifying further:
h = (-5 ± √(25 + 1180)) / 2
h = (-5 ± √1205) / 2
Using a calculator to find the approximate value of √1205, we get:
h ≈ (-5 ± 34.72) / 2
Therefore, the two possible solutions for h are:
h ≈ (-5 + 34.72) / 2 ≈ 14.86
h ≈ (-5 - 34.72) / 2 ≈ -19.86
As we are looking for a positive height, we can ignore the negative value. Therefore, the height that will give the desired area is approximately 14.86 yards.
So the correct answer is:
14.86 yards
To find the height that will give the desired area, we can use the given equation h^2 + 5h = 295.
First, let's rearrange the equation to quadratic form: h^2 + 5h - 295 = 0.
Now, we can use the quadratic formula to solve for h. The quadratic formula is given by:
h = (-b ± sqrt(b^2 - 4ac)) / 2a,
where a, b, and c are the coefficients of the quadratic equation.
In our case, the coefficients are:
a = 1, b = 5, and c = -295.
Substituting these values into the quadratic formula, we get:
h = (-5 ± sqrt(5^2 - 4(1)(-295))) / (2*1).
Simplifying this expression, we have:
h = (-5 ± sqrt(25 + 1180)) / 2.
h = (-5 ± sqrt(1205)) / 2.
Now, let's calculate the possible values of h:
h = (-5 + sqrt(1205)) / 2 ≈ 14.86 yards,
h = (-5 - sqrt(1205)) / 2 ≈ -29.71 yards.
Since the height cannot be negative in this context, we round the height to the nearest hundredth of a yard:
The correct answer is approximately 14.86 yards.