A bridge shaped like a parabolic arch has horizontal distance of 20m. the center and highest point of the arch is 5m above the ground. What is the height of the bridge at a point 5m from the center?
Since the bridge is shaped like a parabolic arch, we can use the equation for a parabola in standard form:
y = a(x - h)^2 + k
where (h, k) is the vertex of the parabola and a is a constant that determines the shape of the parabola. Since the center and highest point of the arch is 5m above the ground, the vertex of the parabola is (0, 5). To find the value of a, we can use the fact that the horizontal distance of the arch is 20m. Since the vertex is at the center, the arch extends 10m to the left and 10m to the right. Therefore, the points on the arch that are 10m to the left and right of the center are (-10, 5) and (10, 5), respectively. Substituting these values into the equation of the parabola, we get:
5 = a(-10 - 0)^2 + 5 and 5 = a(10 - 0)^2 + 5
Simplifying these equations, we get:
100a = 0 and 100a = 0
Since both equations are equal to 0, we can conclude that a = 0. Therefore, the equation of the parabola is simply:
y = 5
This means that the height of the bridge is constant at 5m for all values of x, including x = 5m, which is 5m from the center of the arch. Therefore, the height of the bridge at a point 5m from the center is 5m.
Answers & Comments
Answer and Step-by-step explanation:
Since the bridge is shaped like a parabolic arch, we can use the equation for a parabola in standard form:
y = a(x - h)^2 + k
where (h, k) is the vertex of the parabola and a is a constant that determines the shape of the parabola. Since the center and highest point of the arch is 5m above the ground, the vertex of the parabola is (0, 5). To find the value of a, we can use the fact that the horizontal distance of the arch is 20m. Since the vertex is at the center, the arch extends 10m to the left and 10m to the right. Therefore, the points on the arch that are 10m to the left and right of the center are (-10, 5) and (10, 5), respectively. Substituting these values into the equation of the parabola, we get:
5 = a(-10 - 0)^2 + 5 and 5 = a(10 - 0)^2 + 5
Simplifying these equations, we get:
100a = 0 and 100a = 0
Since both equations are equal to 0, we can conclude that a = 0. Therefore, the equation of the parabola is simply:
y = 5
This means that the height of the bridge is constant at 5m for all values of x, including x = 5m, which is 5m from the center of the arch. Therefore, the height of the bridge at a point 5m from the center is 5m.