This one is tough. "A ballon in the shape of a cylinder with hemispherical ends of the same radius as that of the cylinder is shown. The ballon is being inflated at the rate of 261pi cubic centimeters per minute. At the instant the radius of the cylinder is 3 centimeters, the volume of the ballon is 144pi cubic centimeters and the radius of the cylinder is increasing at the rate of 2 centimeters per minute. At this instant, how fast is the height of the entire ballon increasing?"
Use Newton's Method to find the root of 2x2+5=ex accurate to six decimal places in the interval [3,4].
(Note: Answers should be in decimal form. Up to two decimal places only.)
To find how fast the height of the entire balloon is increasing, we need to calculate the rate of change of the height with respect to time.
Let's denote the height of the cylinder as h and the radius as r. Since the balloon is in the shape of a cylinder with hemispherical ends of the same radius as the cylinder, we can split the balloon into two parts: the cylinder (without the hemispheres) and the two hemispheres.
To find the rate of change of the height, we'll use the chain rule. The volume of the balloon is given by the sum of the volumes of the cylinder and the two hemispheres:
V = V_cylinder + V_hemispheres
The volume of the cylinder is given by V_cylinder = πr^2h, and the volume of the hemispheres is given by V_hemispheres = 2(2/3πr^3) = (4/3)πr^3.
Differentiating both sides of the equation with respect to time (t), we get:
dV/dt = dV_cylinder/dt + dV_hemispheres/dt
Given that the rate of change of volume, dV/dt, is 261π cubic centimeters per minute, and we're trying to find the rate of change of the height, dh/dt, we need to find dV_cylinder/dt and dV_hemispheres/dt.
To express the volume of the cylinder in terms of the height (h), we recognize that the radius (r) of the cylinder is increasing at a rate of 2 centimeters per minute. So, we can express r in terms of h using similar triangles:
r/h = 3/2
Simplifying, we find r = (3/2)h. Substituting this value for r in the formula for the volume of the cylinder, we have:
V_cylinder = π[(3/2)h]^2h = (9/4)πh^3
Differentiating V_cylinder with respect to time using the chain rule, we get:
dV_cylinder/dt = (9/4)π(3h^2)(dh/dt)
Similarly, we can express the volume of the hemispheres in terms of the radius (r). Since the radius is equal to h/2 (since it's half the diameter), we can write:
V_hemispheres = (4/3)π[(h/2)^3] = (1/6)πh^3
Differentiating V_hemispheres with respect to time using the chain rule, we get:
dV_hemispheres/dt = (1/6)π(3h^2)(dh/dt)
Now we can substitute the given values into the equations. At the instant when the radius (r) of the cylinder is 3 centimeters, the volume of the balloon is 144π cubic centimeters. So, we can write:
V_cylinder + V_hemispheres = 144π
Substituting the volume expressions and the values for V_cylinder and V_hemispheres, we have:
(9/4)πh^3 + (1/6)πh^3 = 144π
Simplifying, we get:
(21/12)h^3 = 144
h^3 = (144 * 12) / 21
h^3 = 64
Taking the cube root of both sides, we find:
h = 4 centimeters
Now that we know the current height of the cylinder (h), and we're given that the radius (r) is increasing at a rate of 2 centimeters per minute, we can find dh/dt by substituting the values into the equation for dV_cylinder/dt:
dV_cylinder/dt = (9/4)π(3h^2)(dh/dt)
261π = (9/4)π(3 * 4^2)(dh/dt)
Canceling out the π, we have:
261 = (9/4)(3 * 4^2)(dh/dt)
Simplifying, we find:
261 = 9 * 3 * 16 * (dh/dt)
Solving for dh/dt, we get:
dh/dt = 261 / (9 * 3 * 16)
dh/dt = 261 / 432
Evaluating this expression, we find:
dh/dt ≈ 0.6041 centimeters per minute
Therefore, at the instant the radius of the cylinder is 3 centimeters, the height of the entire balloon is increasing at a rate of approximately 0.6041 centimeters per minute.