An arch is in the form of a semi ellipse.it is 50m wide at the base and has ahight of 20m.how wide is the arch at the height of 10m above the base?
25
since a=25 and b=20, the ellipse can be represented by
x^2/25^2 + y^2/20^2 = 1
Now just find 2x when y=10.
To find the width of the arch at a height of 10m above the base, you can use the equation for the semi-ellipse:
h² = a² - b²,
where h is the height, a is the width at the base, and b is the width at the desired height.
Given:
- Width at the base (a) = 50m
- Height (h) = 20m
Let's substitute these values into the equation to find b:
20² = 50² - b²,
400 = 2500 - b²,
b² = 2500 - 400,
b² = 2100.
Taking the square root of both sides, we get:
b = √2100.
If we substitute this value into a calculator, we find:
b ≈ 45.825m.
Therefore, the width of the arch at a height of 10m above the base is approximately 45.825m.
To determine the width of the arch at a height of 10m above the base, we can use the properties of a semi-ellipse.
A semi-ellipse is defined by its width (the distance between the two endpoints of the base, also called the major axis) and its height (the vertical distance from the base to the highest point of the arch, also called the minor axis).
In this case, we are given that the width of the arch (the major axis) is 50m and the height (the minor axis) is 20m.
To find the width at a specific height, we need to use the equation for a semi-ellipse:
x^2 / a^2 + y^2 / b^2 = 1
where x is the horizontal distance from the center of the semi-ellipse, y is the vertical distance from the center, a is half the width (major axis) of the ellipse, and b is half the height (minor axis) of the ellipse.
In our case, a = 25m (half of 50m) and b = 10m.
To find the width of the arch (x) at a height of 10m above the base (y), we need to rearrange the equation and solve for x:
x = sqrt[(1 - (y^2 / b^2)) * a^2]
Plugging in the values, we have:
x = sqrt[(1 - (10^2 / 10^2)) * 25^2]
= sqrt[(1 - 1) * 25^2]
= sqrt[(0) * 25^2]
= sqrt[0]
= 0
Therefore, at a height of 10m above the base, the width of the arch is 0m. This means that the arch narrows down to a point at this height.