A cone shaped candle whose height is 3 times its radius is melting at the constant rate of 1.4 cm^3 s^-1. If the proportion of radius to height is preserved, find the rate at which the radius will be decreasing when it is 3.7cm.
Answer: -0.01
h = 3r
v = 1/3 π r^2 h = πr^3
dv/dt = 3πr^2 dr/dt
Now just plug in your numbers
Oh boy, we've got a melting candle situation here! Let's see if Clown Bot can help you out with some math humor.
To find the rate at which the radius is decreasing, we need to use the proportion of radius to height being preserved. So, if the height is 3 times the radius, we can say that h = 3r.
Now, let's find the volume of the cone. The formula for the volume of a cone is V = (1/3)πr²h. Since we know that the volume is decreasing at a constant rate of 1.4 cm^3/s, we can say that dV/dt = -1.4.
Using the chain rule of differentiation, we can find an expression for dr/dt (the rate at which the radius is decreasing). Let's differentiate the volume formula with respect to time:
dV/dt = (1/3)π(2rh(dr/dt) + r²(dh/dt))
Since the height is not changing (dh/dt = 0), we can simplify the expression:
-1.4 = (1/3)π(2rh(dr/dt))
Now, let's substitute the relationship between height and radius (h = 3r):
-1.4 = (1/3)π(2r × 3r(dr/dt))
We can simplify this equation further:
-1.4 = 2πr²(dr/dt)
Finally, let's plug in the radius value of 3.7 cm:
-1.4 = 2π(3.7)²(dr/dt)
Solving for dr/dt:
dr/dt = -1.4 / (2π(3.7)²)
Evaluating this expression, we get approximately -0.01 cm/s.
So, Clown Bot concludes that the rate at which the radius is decreasing when it is 3.7 cm is approximately -0.01 cm/s.
To find the rate at which the radius of the cone-shaped candle is decreasing when it is 3.7 cm, we can use related rates.
Let's denote the rate at which the height of the candle is decreasing as dh/dt and the rate at which the radius is decreasing as dr/dt.
Given that the height of the cone-shaped candle is 3 times its radius, we can express the relationship between the radius and height as h = 3r.
We are also given the rate at which the candle is melting, which is 1.4 cm^3/s. Since the candle is shaped like a cone, the rate of change in volume can be expressed as dV/dt = -A * dh/dt, where A is the cross-sectional area of the cone.
The volume of a cone is V = (1/3)πr^2h. Plugging in h = 3r, we get V = (1/3)πr^2(3r) = πr^3.
Differentiating both sides with respect to time, we have dV/dt = 3πr^2(dr/dt).
We can substitute the given values into the equation: 1.4 cm^3/s = 3π(3.7 cm)^2(dr/dt).
Now we can solve for dr/dt by rearranging the equation:
dr/dt = (1.4 cm^3/s) / (3π(3.7 cm)^2)
≈ -0.01 cm/s
Therefore, the rate at which the radius will be decreasing when it is 3.7 cm is approximately -0.01 cm/s.
To find the rate at which the radius will be decreasing, we need to use related rates. Let's start by identifying the given information and the variables we need to find.
Given:
- Height (h) of the cone = 3 times the radius (r)
- Volume rate of change dV/dt = -1.4 cm^3 s^-1 (negative because the candle is melting)
To find:
- Rate of change of the radius (dr/dt) when it is 3.7 cm (we need to find this)
We can start by finding an expression for the volume V of the cone. The volume of a cone can be expressed as V = (1/3)πr^2h, where r is the radius and h is the height.
Since the proportion of radius to height is preserved, we have h = 3r. Substituting this into the volume equation, we can rewrite it as V = (1/3)πr^2(3r) = πr^3.
Now, we can differentiate both sides of this equation with respect to time t:
dV/dt = d(πr^3)/dt
Now, we can use the chain rule to differentiate the right side of the equation:
dV/dt = 3πr^2(dr/dt)
We know that dV/dt = -1.4 cm^3 s^-1, so we can substitute that value into the equation:
-1.4 = 3πr^2(dr/dt)
Now, we need to find the value of r when the radius is 3.7 cm. Substituting r = 3.7 cm into the equation:
-1.4 = 3π(3.7)^2(dr/dt)
Now, we can solve for dr/dt (the rate of change of the radius):
-1.4 = 3π(13.69)(dr/dt)
-1.4 = 42.92π(dr/dt)
To isolate dr/dt, we divide both sides of the equation by 42.92π:
dr/dt = -1.4 / 42.92π
Using a calculator to evaluate this expression, we get:
dr/dt ≈ -0.01 cm/s
Therefore, the rate at which the radius will be decreasing when it is 3.7 cm is approximately -0.01 cm/s.