The Parthenon in Athens, Greece, is an ancient structure that has a rectangular base. The length of the base of the Parthenon is 8 meters more than twice its width. The area of the base is about 2170 square meters. Find the length and width of the base.
L = 2W +8
L * W = (2W+8) + W = 2170
Calculate W, then L.
Step 1: Let's assume the width of the base of the Parthenon is 'W' meters.
Step 2: According to the given information, the length of the base is 8 meters more than twice its width. So, the length can be calculated as: 2W + 8.
Step 3: The area of the base is given as 2170 square meters. We can find the area of a rectangle using the formula: Area = Length * Width.
Step 4: Substituting the values into the formula, we get:
2170 = (2W + 8) * W.
Step 5: Expanding the equation, we get:
2170 = 2W^2 + 8W.
Step 6: Rearranging the equation to make it a quadratic equation, we get:
2W^2 + 8W - 2170 = 0.
Step 7: We can solve this quadratic equation using different methods such as factoring, completing the square, or using the quadratic formula. However, in this case, let's use the quadratic formula:
W = (-b ± √(b^2 - 4ac)) / (2a).
Step 8: Substituting the values into the quadratic formula, we get:
W = (-8 ± √(8^2 - 4*2*(-2170))) / (2*2).
W = (-8 ± √(64 + 17360)) / 4.
W = (-8 ± √(17424)) / 4.
W = (-8 ± 132) / 4.
W = (132 - 8) / 4 or W = (-132 - 8) / 4.
W = 124 / 4 or W = -140 / 4.
W = 31 or W = -35.
Step 9: Since the width cannot be negative, we discard W = -35.
Step 10: The width of the base is 31 meters.
Step 11: Now, we can calculate the length using the formula: Length = 2W + 8.
Length = 2(31) + 8.
Length = 62 + 8.
Length = 70.
Step 12: The length of the base is 70 meters.
So, the length and width of the base of the Parthenon are 70 meters and 31 meters, respectively.
To find the length and width of the base of the Parthenon, we can use algebra.
Let's assume the width of the base is 'x' meters.
According to the given information, the length of the base is 8 meters more than twice the width. So, the length can be represented as 2x + 8.
The area of a rectangle is calculated by multiplying its length and width. In this case, the area of the base is given as 2170 square meters. So, we can set up an equation:
x * (2x + 8) = 2170
Now, let's solve the equation to find the width 'x' and the length '2x + 8'.
Expanding the equation:
2x^2 + 8x = 2170
Rearranging the equation to form a quadratic equation:
2x^2 + 8x - 2170 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. If factoring is not immediately obvious, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
For the equation 2x^2 + 8x - 2170 = 0, the values of a, b, and c are:
a = 2
b = 8
c = -2170
Using the quadratic formula:
x = (-8 ± √(8^2 - 4 * 2 * -2170)) / (2 * 2)
Simplifying and calculating the roots:
x = (-8 ± √(64 + 17360)) / 4
x = (-8 ± √(17424)) / 4
x = (-8 ± 132) / 4
Now, we can calculate the two possible values of x:
x1 = (-8 + 132) / 4 = 124 / 4 = 31
x2 = (-8 - 132) / 4 = -140 / 4 = -35
Since width cannot be negative, we discard the value of x = -35.
Therefore, the width of the base of the Parthenon is 31 meters.
Now, we can find the length by substituting the value of x into the expression 2x + 8:
length = 2 * 31 + 8 = 62 + 8 = 70
Therefore, the length of the base of the Parthenon is 70 meters, and the width is 31 meters.