help on application of differentaion please?

Question

1. Find the equations of the 2 tangents to the ellipse x^2+4y^2=8 which are parallel to the line x+2y=6

2. Water is poured into an inverted conical container of base radius 5m and height 15m at a rate of 12m^3 min^-1. Calculate rate at which the water level is rising when radius of the water surface is 2m.

Answer 1

1. The given line, x + 2y = 6, can be rewritten as y = -0.5x + 3, so we can see that it has slope of -0.5. Thus, we need to find the points at which the line tangent to the ellipse has slope = -0.5.

First, I would separate the equation of the ellipse into two functions: y = sqrt(2 - 0.25x^2) and y = -sqrt(2 - 0.25x^2). Now differentiate to get dy/dx = -0.25x / 2sqrt(2 - 0.25x^2) and dy/dx = 0.25x / 2sqrt(2 - 0.25x^2). Now we must solve for these expressions equal to -0.5. I'll start with the first (positive) function.

-0.25x / 2sqrt(2 - 0.25x^2) = -0.5
-0.25x = -sqrt(2 - 0.25x^2)
x = 4sqrt(2 - 0.25x^2)
x^2 = 16(2 - 0.25x^2)
x^2 = 32 - 4x^2
5x^2 = 32
x = +/- sqrt(32/5)

However, only the positive value satisfies the equation. Conveniently, though, x = -sqrt(32/5) satisfies dy/dx = -0.5 for the second (negative) function.

I leave it for you to use these values for x to solve for y (don't forget to use the function for the correct half of the ellipse, and check your answers on +/- results to make sure you have a valid solution). You'll then have two ordered pairs, (sqrt(32/5), y1) and (-sqrt(32/5), y2). From those, you can write the slope-intercept form of a line, which is y - y1 = m(x - x1), where (x1, y1) is a point on the line, and m is the slope (which you know is -0.5).

2. This is a related rates problem. Start with cone geometry. The volume of a cone is V = (1/3)pi(r^2)h, where r is radius and h is height. As water is poured in, the cone of water increases in height and its radius increases proportionally. In this cone, we can say r = h/3 because we know that the base radius is 5 and the height is 15, so r/h = 5/15 = 1/3. This means the volume of the cone can be written as V = (1/3)pi((h/3)^2)h = (1/3)pi(h^3)/9 = (1/27)pi*h^3.

Now, differentiate both sides with respect to time, using the chain rule on the right hand side, to get dV/dt = (1/9)pi(h^2)(dh/dt). You have been given dV/dt = (12 m^3/min). Knowing r = 2 m allows you to find h = 3r = 6 m. So now your only unknown is dh/dt, which is what you are trying to find ("the rate at which the water level is rising"); you can solve for it algebraically.

Answer 2

well i can answer 1 for you .. if you differentiate it you get
2x + 8y = 8
x + 2y = 6

then do simultaneous equations (hint hint) :)