A fireplace arch is to be constructed in the form of a semiellipse. 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 semiellipse.
a.) What is the total length of the string?
b.) How far from the center should the string be nailed into the base?
To find the total length of the string and the distance from the center to the nails, we can use properties of the ellipse.
a.) To calculate the total length of the string, we need to find the length of the semiellipse's arc.
The formula to calculate the arc length of the ellipse in terms of its semi-major and semi-minor axes, a and b respectively, is given as:
Arc Length = π(a + b) [1 + (3h^2) / (10(a - b))^2] / 2a
In this case, the semi-major axis of the ellipse is 4 feet (half of the width along the base) and the semi-minor axis is 3 feet (half of the height at the center). The height at the center can also be considered as the distance from the center to the topmost point of the ellipse.
Using these values in the formula, we get:
Arc Length ≈ π(4 + 3) [1 + (3 * 3^2) / (10(4 - 3))^2] / (2 * 4)
Arc Length ≈ π(7) [1 + (27) / (10)^2] / 8
Arc Length ≈ π(7) [1 + 27 / 100] / 8
Arc Length ≈ π(7) [ (100 + 27) / 100 ] / 8
Arc Length ≈ π(7) (127 / 100) / 8
Arc Length ≈ 13.241 feet (rounded to three decimal places)
Therefore, the total length of the string that needs to be cut is approximately 13.241 feet.
b.) To determine the distance from the center to the nails along the base, we need to find the distance along the vertical axis where the string is attached.
This can be calculated using the equation of an ellipse, which relates the semi-major axis (a), semi-minor axis (b), and the height (h) at any point on the ellipse.
The equation for the ellipse is:
b^2 = a^2 - (a^2 * h^2) / a^2
In this case, we can substitute the known values: a = 4 and h = 3 (height at the center).
b^2 = 4^2 - (4^2 * 3^2) / 4^2
b^2 = 16 - (16 * 9) / 16
b^2 = 16 - 144 / 16
b^2 = 16 - 9
b^2 = 7
Taking the square root of both sides, we find:
b = sqrt(7)
b ≈ 2.646 feet (rounded to three decimal places)
Therefore, the string should be nailed into the base at a distance of approximately 2.646 feet from the center.
recall that an ellipse with semiaxes a and b and focus at c has the property that
a^2 = b^2 + c^2
The sum of the distances from (x,y) on the ellipse is 2a.