A propane tank is in the shape generated by revolving the region enclosed by the right half of the graph of
(4*x^2)+(64*y^2)=144 and the y-axis
about the y-axis. If x and y are measured in meters find the depth of the propane in the tank when it is filled to one-quarter of the tank's volume. (Round to 3 decimal places).
HINT: You will need to use a Graphing Calculator, Online Graphing Calculator, or Computer Algebra System to solve an equation towards the end of your solution to this problem.
To find the depth of the propane in the tank when it is filled to one-quarter of the tank's volume, we need to follow these steps:
Step 1: Understand the problem and the given information.
We are given the equation of the graph that represents the shape of the propane tank. The equation is (4*x^2) + (64*y^2) = 144. Here, x and y are measured in meters.
Step 2: Find the volume of the propane tank.
To find the volume of the propane tank, we need to calculate the definite integral of the function generated by revolving the region enclosed by the graph and the y-axis around the y-axis. The integral represents the volume under the curve.
The equation (4*x^2) + (64*y^2) = 144 can be rearranged to solve for x:
4*x^2 = 144 - 64*y^2
x^2 = (144 - 64*y^2) / 4
x = sqrt[(144 - 64*y^2) / 4]
To get the volume, we integrate the function [(144-64y^2)/4]πy^2 with respect to y, from 0 to the upper bound, which we need to find.
Step 3: Find the upper bound of the integral.
To find the upper bound of the integral, we need to determine the y-coordinate where the tank is filled to one-quarter of its volume. Let's assume this value is y = d, where d represents the depth in meters.
Step 4: Solve for the upper bound.
Now, we need to find the value of y such that the volume of the propane tank, when filled to y = d, is equal to one-quarter of the total volume.
To solve this equation, we need a graphing calculator, online graphing calculator, or a computer algebra system, as mentioned in the hint. By graphing the functions V(d) = ∫[0 to d][{(144-64y^2)/4}πy^2] dy and 1/4 of the total volume, we can find the intersection point(s). The x-coordinate of the intersection point will give us the value of d.
Step 5: Calculate the depth of the propane in the tank.
Using a graphing calculator or a computer algebra system, graph the function V(d) = ∫[0 to d][(144-64y^2)/4]πy^2 dy and the function representing one-quarter of the total volume. Find the intersection point(s). The x-coordinate of the intersection point will give us the value of d, the depth of the propane in the tank.
Round the value of d to 3 decimal places to get the final answer.