A fireplace arch is to be constructed in the form of a semi ellipse. The opening is to have a height of 3 feet at the center and a width of 8 feet along the base. The contractor cuts a string of a certain length and nails each end of the string along the base in order to sketch the outline of the semi ellipse.
a.) What is the total length of the string?
b.)How far from the center should the string be nailed into the base?
He must nail the ends of the string at the focal points of the ellipse
For this ellipse:
c^2 + 3^2 = 4^2
c^2 = 7
c = √7
so one of the focal points is (√7, 0) , clear √7 units form the centre
using the definition of ellipse properties, the length of the string is 8 units
a.) Well, the total length of the string used can be found by using the formula for the circumference of a semi ellipse. Let's call the length of the string "L". To find it, we can use the following formula: L = π × (3 + (8/2)). So, the total length of the string is L = π × (3 + 4) = π × 7.
b.) Now, to determine how far from the center the string should be nailed into the base, we need to find the distance between the center and the base of the semi ellipse. Since the height at the center is 3 feet, the distance we're looking for is exactly half of the width, which is 4 feet. So, the string should be nailed into the base at a distance of 4 feet from the center. However, I must warn you, placing a nail in someone's base might not be considered polite.
To find the total length of the string, we can use the formula for the circumference of an ellipse:
Circumference = π * (3 * (a + b) - √((3 * a + b) * (a + 3 * b)))
Where a is the semi-major axis and b is the semi-minor axis of the ellipse.
In this case, the height at the center of the opening is given as 3 feet, which is the semi-minor axis (b). The width along the base is given as 8 feet, which is twice the semi-major axis (2 * a).
a.) Total length of the string = Circumference
= π * (3 * (2 * a + b) - √((3 * a + b) * (a + 3 * b)))
Substituting the given values:
b = 3 feet
a = 8 feet / 2 = 4 feet
Total length of the string = π * (3 * (2 * 4 + 3) - √((3 * 4 + 3) * (4 + 3)))
= π * (3 * (8 + 3) - √((12 + 3) * (4 + 3)))
= π * (3 * 11 - √(15 * 7))
≈ π * (33 - √105)
≈ 97.961 feet (rounded to 3 decimal places)
b.) To find the distance from the center at which the string should be nailed into the base, we can use the formula:
Distance = a^2 / b
Substituting the given values:
a = 4 feet
b = 3 feet
Distance = 4^2 / 3
= 16 / 3
≈ 5.333 feet (rounded to 3 decimal places)
So, the string should be nailed into the base approximately 5.333 feet away from the center.
To find the total length of the string, you need to find the perimeter of the semi-ellipse.
a.) The formula for the circumference of a semi-ellipse is given by the equation:
C = π(a+b),
where "a" is the length of the semi-major axis (half of the width along the base), and "b" is the length of the semi-minor axis (half of the height at the center).
In this case, the width along the base is 8 feet, so the length of the semi-major axis (a) is 8/2 = 4 feet. The height at the center is 3 feet, so the length of the semi-minor axis (b) is 3 feet.
Using the formula, we can calculate the circumference:
C = π(4 + 3) = 7π feet.
Thus, the total length of the string should be approximately 22.0 feet, since π is approximately 3.14.
b.) To find how far from the center the string should be nailed into the base, you need to find the distance from the center of the ellipse to the major axis, also known as the focus.
The formula to calculate the distance (d) from the center to the focus is given by:
d = √((a^2 - b^2)/2),
where "a" is the length of the semi-major axis and "b" is the length of the semi-minor axis.
In this case, a = 4 feet and b = 3 feet.
Using the formula, we can calculate:
d = √((4^2 - 3^2)/2)
= √((16 - 9)/2)
= √(7/2)
= √3.5 feet.
Thus, the string should be nailed into the base approximately 3.5 feet away from the center in order to sketch the outline of the semi-ellipse.