A group of engineers is building a parabolic satellite dish whose shape will be formed by rotating the curve y=ax2 about the y-axis. If the dish is to have a 8-foot diameter and a maximum depth of 2 feet, find the value of a and the surface area (in square feet) of the dish.
89.03
To find the value of "a," we need to use the information given: the dish has a diameter of 8 feet and a maximum depth of 2 feet.
Since the dish is formed by rotating the curve y=ax^2 about the y-axis, we can use the standard equation for a parabola: y = ax^2.
First, let's consider the diameter. The diameter of the dish is 8 feet, which means it has a radius of 4 feet (diameter = 2 * radius).
At the maximum depth, the curve should reach a value of 2 feet. So, we can substitute this value into the equation:
2 = a * 4^2
Simplifying,
2 = 16a
Now, solve for "a" by dividing both sides of the equation by 16:
a = 2/16 = 1/8
Therefore, the value of "a" is 1/8.
Next, let's calculate the surface area of the dish.
The surface area can be found by rotating the curve 2π times around the y-axis. To do this, we'll integrate the function 2πy*dx from x = -4 to x = 4.
Using the equation y = ax^2 with a = 1/8, we have:
y = (1/8)x^2
To calculate the surface area, evaluate the integral:
Surface area = ∫(from -4 to 4) 2π * y * dx
= 2π * ∫(from -4 to 4) (1/8)x^2 * dx
= 2π(1/8) * ∫(from -4 to 4) x^2 * dx
= π/4 * ∫(from -4 to 4) x^2 * dx
To integrate x^2, use the power rule for integration:
∫x^n dx = (1/(n+1)) * x^(n+1) + C
Applying this rule:
= π/4 * [(1/3) * x^3] (from -4 to 4)
= π/4 * (1/3) * [(4^3) - (-4^3)]
= π/4 * (1/3) * [64 - (-64)]
= π/4 * (1/3) * [64 + 64]
= π/4 * (1/3) * 128
= π/4 * 128/3
Simplifying,
= (32π)/3
Therefore, the surface area of the dish is (32π)/3 square feet.
To find the value of 'a' and the surface area of the dish, we need to understand the properties of a parabolic curve and how it relates to a satellite dish.
1. Understanding the Parabolic Curve:
- A parabola is a U-shaped curve that can be defined by the equation y = ax^2, where 'a' is a constant.
- The vertex of the parabola is the point where it "opens" and changes direction.
- When the parabola opens upwards, the vertex is the lowest point on the curve. This is the case with a satellite dish.
2. Determining the value of 'a':
- We are given that the maximum depth of the satellite dish is 2 feet.
- In the equation y = ax^2, the maximum depth corresponds to the y-coordinate of the vertex.
- The vertex is located at the point (0, a) because it lies on the y-axis.
- Therefore, we can conclude that the vertex of the parabolic curve is (0, a).
- Given that the maximum depth is 2 feet, we have a = 2.
3. Finding the surface area of the dish:
- To find the surface area of the dish, we need to calculate the area of the cross-section and then rotate it around the y-axis to form a 3D shape.
- The cross-section of the dish is a circle with a radius equal to half of the diameter.
- Given that the diameter is 8 feet, the radius is 8/2 = 4 feet.
- The equation that represents the circle is x^2 + y^2 = r^2, where r is the radius.
- We need to find the limits of integration. Since the vertex is at (0, 2), we can take the limits from -2 to 2.
- To find the surface area, we will integrate the circumference of the circle from -2 to 2 and then multiply it by 2π.
4. Calculating the surface area:
- The equation of the circle is x^2 + y^2 = 4^2, which simplifies to x^2 + y^2 = 16.
- We need to express x in terms of y to set up the integral.
- Solving for x, we get x = ±√(16 - y^2).
- The limits of integration for y are from -2 to 2. So, we integrate the circumference of the circle from -2 to 2: 2∫[-2 to 2] 2π√(16 - y^2) dy.
- Evaluating this integral will give us the surface area of the dish.
Therefore, the value of 'a' is 2 and the surface area of the dish can be found by evaluating the integral 2∫[-2 to 2] 2π√(16 - y^2) dy.
you can find useful formulas here. Pick the one for a paraboloid and use your numbers.
http://www.had2know.com/academics/paraboloid-surface-area-volume-calculator.html
I'm sure you can derive it by calculating the arc length and revolving the curve. The Theorem of Pappus might help.