The volume of a cube is increasing at the rate of 20 cubic centimeters per second. How fast, in square centimeters per second, is the surface area of the cube increasing at the instant when each edge of the cube is 10 centimeters long?
Consider that the edge of the cube varies with time so that is a function of time
By signing up, you agree to our Terms of Service and Privacy Policy
To find the rate at which the surface area of the cube is increasing when each edge is 10 centimeters long, we'll use the formula for the surface area of a cube, which is ( 6s^2 ), where ( s ) is the length of the side of the cube.
Given that the volume of the cube is increasing at a rate of 20 cubic centimeters per second, we can use the formula for the volume of a cube, which is ( s^3 ), where ( s ) is the length of the side of the cube.
Differentiating the volume formula with respect to time ( t ), we get: [ \frac{dV}{dt} = 3s^2 \frac{ds}{dt} ]
Given that ( \frac{dV}{dt} = 20 ) cubic centimeters per second, and when ( s = 10 ), we can find ( \frac{ds}{dt} ): [ 20 = 3(10)^2 \frac{ds}{dt} ] [ 20 = 300 \frac{ds}{dt} ] [ \frac{ds}{dt} = \frac{20}{300} = \frac{1}{15} ] centimeters per second.
Now, using the formula for the surface area of the cube, when ( s = 10 ): [ A = 6(10)^2 = 600 ] square centimeters.
Differentiating the surface area formula with respect to time ( t ), we get: [ \frac{dA}{dt} = 12s \frac{ds}{dt} ]
Substituting the values ( s = 10 ) and ( \frac{ds}{dt} = \frac{1}{15} ) into the formula: [ \frac{dA}{dt} = 12(10) \left( \frac{1}{15} \right) ] [ \frac{dA}{dt} = \frac{8}{5} ] square centimeters per second.
Therefore, the surface area of the cube is increasing at a rate of ( \frac{8}{5} ) square centimeters per second when each edge of the cube is 10 centimeters long.
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
- How do you find the dimensions of the aquarium that minimize the cost of the materials if the base of an aquarium with volume v is made of slate and the sides are made of glass and the slate costs five times as much (per unit area) as glass?
- How do you find the linearization of #f(x)=lnx # at x=8?
- A street light is at the top of a 16 ft tall pole. A woman 6 ft tall walks away from the pole with a speed of 8 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 35 ft from the base of the pole?
- If the rate of change in #x# is #"3 s"^(-1)#, and #(dy)/(dx) = 5#, what is the rate of change in #y#? Is #y# changing faster than #x# or vice versa?
- How many seconds will the ball be going upward if a ball is thrown vertically upward from the ground with an initial velocity of 20 feet per second and #a(t)= -32# feet per second squared?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7