Problem 91 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

CONSERVATION A lake has roughly the same shape as the bottom half of the solid formed by rotating the curve \(2 x^{2}+3 y^{2}=6\) about the \(x\) axis, for \(x\) and \(y\) measured in miles. Conservationists want the lake to contain 1,000 trout per cubic mile. If the lake currently contains 5,000 trout, how many more must be added to meet this requirement?

Step-by-Step Solution

Answer

Approximately 22,095 trout must be added.

Step 1: Identify the Equation of the Solid

The equation of the curve given is \[2x^2 + 3y^2 = 6\]. This will be rotated about the \(x\)-axis to form the solid.

Step 2: Express in Standard Form

Rewrite the equation as \[y^2 = \frac{6 - 2x^2}{3}\].

Step 3: Set Up the Volume Integral

To find the volume of the solid obtained by rotating the curve about the \(x\)-axis, use the volume of revolution integral \[V = \pi \int_{a}^{b} [f(x)]^2 dx\]. Here, \[f(x) = \sqrt{\frac{6 - 2x^2}{3}}\].

Step 4: Determine the Limits of Integration

Solve \[2x^2 + 3y^2 = 6\] for when \(y = 0\) to find the range of x. This gives \[2x^2 = 6\] hence, \[x = \pm \sqrt{3}\]. So the limits of integration are from \[-\sqrt{3}\] to \[\sqrt{3}\].

Step 5: Calculate the Integral

Substitute \[f(x)\] into the integral: \[V = \pi \int_{-\sqrt{3}}^{\sqrt{3}} \left( \sqrt{\frac{6 - 2x^2}{3}} \right)^2 dx\]. This simplifies to \[V = \pi \int_{-\sqrt{3}}^{\sqrt{3}} \frac{6 - 2x^2}{3} dx\].

Step 6: Simplify and Solve

Integrate and solve this: \[V = \pi \int_{-\sqrt{3}}^{\sqrt{3}} \frac{6}{3} - \frac{2x^2}{3} dx\] \[V = \pi \int_{-\sqrt{3}}^{\sqrt{3}} 2 - \frac{2x^2}{3} dx\]. Integrate to get \[V = \pi \left[ 2x - \frac{2x^3}{9} \right]_{-\sqrt{3}}^{\sqrt{3}}\]. Evaluate this at the limits to get \[V = \pi \left[ 2\sqrt{3} - \frac{2(\sqrt{3})^3}{9} - (2(-\sqrt{3}) - \frac{2(-\sqrt{3})^3}{9}) \right] \]. Simplify to get the final volume: \[V = \frac{16\pi\sqrt{3}}{9}\] cubic miles.

Step 7: Calculate the Trout Required

The volume of the lake is \[ \frac{16\pi\sqrt{3}}{9} \] cubic miles and each cubic mile should contain 1,000 trout. Thus, the total number of trout required is \[1000 \times \frac{16\pi\sqrt{3}}{9} = \frac{16000\pi\sqrt{3}}{9} \].

Step 8: Determine the Additional Trout Needed

The lake currently contains 5,000 trout. The difference \[ \frac{16000\pi\sqrt{3}}{9} - 5000 \] gives the number of trout to be added.

Step 9: Approximate the Final Answer

Calculating the numeric approximation for the exact requirement: \[ \frac{16000\pi\sqrt{3}}{9} - 5000 \approx 27095.1 - 5000 = 22095.1 \] trout must be added.

Key Concepts

calculus

Calculus is the mathematical study of change. It helps us understand how quantities change and accumulate over time. In the given exercise, we used calculus to determine the volume of a solid formed by rotating a curve around an axis, also known as a solid of revolution. To solve this type of problem, we perform integration, another key concept of calculus.

Calculus enables us to find the area under a curve, the rate of change at any given point, and as in this problem, the volume of complex shapes. The solid of revolution concept specifically involves integrating to sum up an infinite number of infinitesimally small slices of the solid to find its volume.

Understanding both differential and integral calculus is crucial for addressing such problems. Here, we primarily use integral calculus to handle the rotation of the curve.

integration

Integration is a fundamental concept in calculus. It allows us to calculate areas, volumes, and other quantities when they are described by curves and surfaces. In the exercise, we used an integral to find the volume of the lake formed by rotating the given curve around the x-axis.

Specifically, the volume of revolution formula was used:



V =
π ∫[ a b ]
[ f( x ) ]² dx
.

This formula sums up the volumes of infinitesimally thin disks, each of which has a radius given by the function f(x) and thickness dx.

In our problem, we first squared f( x ) =

√( 6 - 2x² ) /3

,
giving us V =
π ∫[ -√3 √3 ]
( 6 - 2x² )/3 dx

as the integral to solve.

solid of revolution

A solid of revolution is a three-dimensional shape created by rotating a two-dimensional curve around an axis. To visualize this, imagine a curve on a flat piece of paper. When you spin the paper around a line (the axis), the curve sweeps out a volume in space, creating a solid shape.

In the problem, we started with the equation 2x² + 3y² = 6 ,

representing a curve. By rotating this around the x-axis, we formed a three-dimensional lake shape. To find out how many trout fit in this lake, we needed to calculate the volume of this solid.

Using the concept of a solid of revolution, we set up an integral by expressing y in terms of x and integrating this area over the interval from -√3 to √3 . This process gave us the volume of the lake, which we then used to determine the necessary number of trout.