The area of a trapezoid is 66 sq units. The length of its longer base is 4 units longer than the length of its shorter base, and its height is 7 units longer than the length of its shorter base. Find the length of each base and the height of the trapezoid.
you know that with bases b and c, and height h, a trapezoid has area
a = h * (b+c)/2
Just plug in your numbers, letting x be the length of the short base:
a = (x+7) * (x + x+4)/2
66 = (x+7)(x+2)
x^2 + 9x - 52 = 0
(x+13)(x-4) = 0
x = 4
short base = 4
long base = 8
height = 11
area = 11(8+4)/2 = 11*6 = 66
Let's assume the length of the shorter base of the trapezoid as 'x' units.
According to the given information, the longer base is 4 units longer than the shorter base. Therefore, the length of the longer base can be expressed as 'x + 4' units.
The height of the trapezoid is 7 units longer than the length of the shorter base. Hence, the height can be represented as 'x + 7' units.
The formula for finding the area of a trapezoid is given by:
Area = (1/2) * (sum of bases) * height
Substituting the given values, we get:
66 = (1/2) * (x + (x + 4)) * (x + 7)
To simplify the equation, let's solve the expression inside the parentheses:
66 = (1/2) * (2x + 4) * (x + 7)
66 = (x + 2) * (x + 7)
Expanding the equation:
66 = x^2 + 7x + 2x + 14
66 = x^2 + 9x + 14
Rearranging the equation to make it a quadratic equation:
x^2 + 9x + 14 - 66 = 0
x^2 + 9x - 52 = 0
To solve this quadratic equation, we can use factorization or the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = 9, and c = -52:
x = (-9 ± √(9^2 - 4 * 1 * -52)) / (2 * 1)
x = (-9 ± √(81 + 208)) / 2
x = (-9 ± √289) / 2
Simplifying the equation further:
x = (-9 ± 17) / 2
This gives two possible values for 'x':
1) x = (-9 + 17) / 2 = 8 / 2 = 4
2) x = (-9 - 17) / 2 = -26 / 2 = -13
Since the length of a side cannot be negative, we discard the second solution.
Hence, the length of the shorter base of the trapezoid is 4 units.
The length of the longer base can be calculated by adding 4 to the length of the shorter base:
Length of longer base = 4 + 4 = 8 units.
The height of the trapezoid can be found by adding 7 to the length of the shorter base:
Height = 4 + 7 = 11 units.
Therefore, the length of the shorter base is 4 units, the length of the longer base is 8 units, and the height is 11 units for the given trapezoid.
To find the length of each base and the height of the trapezoid, let's denote the shorter base as 'x' units.
According to the given information, the longer base is 4 units longer than the shorter base, which means the longer base is (x + 4) units.
The height of the trapezoid is 7 units longer than the shorter base, so the height is (x + 7) units.
The formula for the area of a trapezoid is:
Area = (1/2) * (sum of bases) * height
Given that the area of the trapezoid is 66 sq units, we can substitute the values in the formula:
66 = (1/2) * [(x + x + 4)] * (x + 7)
Simplifying this equation:
66 = (1/2) * (2x + 4) * (x + 7)
66 = (x + 2) * (x + 7)
Expanding the equation:
66 = x^2 + 7x + 2x + 14
66 = x^2 + 9x + 14
Rearranging the equation to bring all terms to one side:
x^2 + 9x + 14 - 66 = 0
x^2 + 9x - 52 = 0
To solve this quadratic equation, we can use factoring or the quadratic formula. In this case, factoring is more suitable. We need to find two numbers whose sum is 9 and whose product is -52. These numbers are 13 and -4.
Therefore, the equation factors to:
(x - 4)(x + 13) = 0
Setting each factor to zero:
x - 4 = 0 or x + 13 = 0
Solving both equations:
x = 4 or x = -13
Since we are dealing with lengths, a negative value doesn't make sense in this context. Therefore, the shorter base, x, is 4 units.
The longer base is 4 units longer than the shorter base, so it is (4 + 4) = 8 units.
The height is 7 units longer than the shorter base, so it is (4 + 7) = 11 units.
In conclusion, the length of the shorter base is 4 units, the length of the longer base is 8 units, and the height of the trapezoid is 11 units.