An arch is in the form of a Semi-Ellipse. It is 40 feet wide at the base and has a height of 18 feet. How wide is the
arch at a height of 15 feet above the base? Nearest tenth.
so you have
a = 20
b = 18
so the equation is
x^2/400 + y^2/324 = 1
When y=15, x = 10√11/3
so the arch is 20√11/3 wide at that point
To find the width of the arch at a height of 15 feet above the base, you can use the equation of a semi-ellipse:
(x/a)^2 + (y/b)^2 = 1,
where a is the width of the semi-ellipse at the base and b is the height of the semi-ellipse.
In this case, a = 40 feet and b = 18 feet.
To find the width at a height of 15 feet above the base, we need to find x when y = 15.
So let's solve the equation for x:
(x/40)^2 + (15/18)^2 = 1.
Multiply both sides of the equation by (40^2 * 18^2):
18^2 * x^2 + 40^2 * 15^2 = 40^2 * 18^2.
Simplify:
x^2 = (40^2 * 18^2 - 15^2 * 40^2) / 18^2.
Calculate and simplify:
x^2 = (1600 * 324 - 225 * 1600) / 324,
x^2 = 518400 - 36000 / 324,
x^2 = 482400 / 324,
x^2 = 1488.
Take the square root of both sides to solve for x:
x = √1488.
x ≈ 38.5.
Therefore, the width of the arch at a height of 15 feet above the base is approximately 38.5 feet.
To find the width of the arch at a height of 15 feet above the base, we can use the equation of a semi-ellipse.
The equation of a semi-ellipse is given by:
x^2/a^2 + y^2/b^2 = 1
Where:
a = half of the width of the base of the semi-ellipse
b = height of the semi-ellipse
In this case, we are given that the base width of the semi-ellipse is 40 feet (which means a = 20 feet) and the height of the semi-ellipse is 18 feet (which means b = 18 feet).
Now, we can substitute these values into the equation and solve for x when y = 15 feet:
x^2/20^2 + 15^2/18^2 = 1
x^2/400 + 225/324 = 1
x^2/400 + 25/36 = 1
Multiply both sides of the equation by 400 to eliminate the fractions:
x^2 + (25/36) * 400 = 400
x^2 + (25/36) * 400 = 400
x^2 + 277.77 ≈ 400
Subtract 277.77 from both sides:
x^2 ≈ 122.23
To find the value of x, we can take the square root of both sides:
x ≈ √(122.23)
x ≈ 11.06
Therefore, the width of the arch at a height of 15 feet above the base is approximately 11.1 feet (rounded to the nearest tenth).