How fast is the surface area of a cube changing when the edge is 30 cm. If the edge is changing at the rate of 3 mm per second?
Well, if the edge of the cube is changing at a rate of 3 mm per second, I must say, it's quite edgy! However, let's get to the math part.
The surface area of a cube is given by the formula A = 6s^2, where s is the length of an edge. So, when the edge is 30 cm, the surface area is 6(30)^2 = 5400 cm^2.
To find how fast the surface area is changing, we'll need to take the derivative of the surface area function with respect to time (t). Since s is changing with respect to t, we'll use the chain rule.
Differentiating both sides with respect to t, we get:
dA/dt = 12s * ds/dt
Plugging in the values, we have:
dA/dt = 12(30 cm) * (3 mm/s)
= 1080 cm^2/s
So, the surface area of the cube is changing at a rate of 1080 cm^2 per second. That's quite the dynamic cube!
To find the rate of change of the surface area of a cube, we can differentiate the formula for the surface area of a cube with respect to time.
The formula for the surface area of a cube is A = 6s^2, where A is the surface area and s is the length of the edge.
Given that the edge is changing at a rate of 3 mm per second, we need to convert this to centimeters per second since the given edge length is in centimeters.
Since 1 cm = 10 mm, the rate of change of the edge in centimeters per second is 3 mm/s รท 10 = 0.3 cm/s.
Next, we can differentiate the formula for the surface area of a cube with respect to time (t):
dA/dt = d/dt (6s^2)
To determine how fast the surface area is changing when the edge is 30 cm, we substitute s = 30 cm into the formula and differentiate:
dA/dt = d/dt (6(30)^2)
Simplifying:
dA/dt = d/dt (6 * 900)
= 0
Therefore, when the edge is 30 cm and changing at a rate of 3 mm per second, the rate of change of the surface area of the cube is 0 cm^2 per second.
To find how fast the surface area of a cube is changing, we need to find the derivative of the surface area formula with respect to time. The surface area (S) of a cube is given by the formula S = 6 * (edge)^2.
Given that the edge is changing at a rate of 3 mm per second, we need to convert this rate to cm per second since the edge is given in cm. There are 10 mm in 1 cm, so the rate of change of the edge is 3 mm/s * (1 cm / 10 mm) = 0.3 cm/s.
Now, let's differentiate the surface area formula with respect to time t: dS/dt = d/dt(6 * (edge)^2).
Using the power rule of differentiation, d/dt(a * x^n) = a * n * x^(n-1), where a is a constant, x is a variable, and n is a constant exponent, we can differentiate the surface area formula:
dS/dt = d/dt(6 * (edge)^2)
= 6 * d/dt((edge)^2)
= 6 * 2 * (edge) * d(edge)/dt
= 12 * (edge) * d(edge)/dt
Now, plug in the given values: edge = 30 cm and d(edge)/dt = 0.3 cm/s.
dS/dt = 12 * (30 cm) * (0.3 cm/s)
= 360 cm^2/s
Therefore, the surface area of the cube is changing at a rate of 360 cm^2/s when the edge is 30 cm and the edge is changing at a rate of 3 mm/s.
A = 6s^2
dA/dt = 12s ds/dt
dA/dt = 12*300*3 mm^2/s