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The volume of a cube is increasing at the rate of 18 cm3 per second. When the edge of the cube is 12 cm, then the rate in cm2/s at which the surface area of the cube increases, is
Correct answer is '6'. Can you explain this answer?
Most Upvoted Answer
The volume of a cube is increasing at the rate of 18 cm3 per second. ...
Given Data:
- Volume of the cube is increasing at the rate of 18 cm3 per second.
- The edge of the cube is 12 cm.
To Find:
- The rate at which the surface area of the cube increases in cm2/s.
Formula:
- The volume of a cube is given by V = a3, where a is the length of the edge.
- The surface area of a cube is given by A = 6a2.
Explanation:
Step 1: Find the rate of change of edge length (da/dt)
- We are given that the volume of the cube is increasing at the rate of 18 cm3 per second.
- Since the volume of a cube is given by V = a3, we can differentiate both sides with respect to time to find the rate of change of the edge length (da/dt).
- dV/dt = 3a2 * da/dt
- Given dV/dt = 18 cm3/s and a = 12 cm, we can solve for da/dt.
Step 2: Find the rate of change of surface area (dA/dt)
- The surface area of a cube is given by A = 6a2.
- We can differentiate both sides with respect to time to find the rate of change of the surface area (dA/dt).
- dA/dt = 12a * da/dt
- Given da/dt from step 1, we can solve for dA/dt.
Step 3: Calculate the rate at which the surface area of the cube increases
- Plug in the values for a and da/dt into the formula dA/dt = 12a * da/dt.
- Given a = 12 cm and da/dt = 0.25 cm/s (from step 1), we can calculate dA/dt.
Final Answer:
- The rate at which the surface area of the cube increases is 6 cm2/s.
- Volume of the cube is increasing at the rate of 18 cm3 per second.
- The edge of the cube is 12 cm.
To Find:
- The rate at which the surface area of the cube increases in cm2/s.
Formula:
- The volume of a cube is given by V = a3, where a is the length of the edge.
- The surface area of a cube is given by A = 6a2.
Explanation:
Step 1: Find the rate of change of edge length (da/dt)
- We are given that the volume of the cube is increasing at the rate of 18 cm3 per second.
- Since the volume of a cube is given by V = a3, we can differentiate both sides with respect to time to find the rate of change of the edge length (da/dt).
- dV/dt = 3a2 * da/dt
- Given dV/dt = 18 cm3/s and a = 12 cm, we can solve for da/dt.
Step 2: Find the rate of change of surface area (dA/dt)
- The surface area of a cube is given by A = 6a2.
- We can differentiate both sides with respect to time to find the rate of change of the surface area (dA/dt).
- dA/dt = 12a * da/dt
- Given da/dt from step 1, we can solve for dA/dt.
Step 3: Calculate the rate at which the surface area of the cube increases
- Plug in the values for a and da/dt into the formula dA/dt = 12a * da/dt.
- Given a = 12 cm and da/dt = 0.25 cm/s (from step 1), we can calculate dA/dt.
Final Answer:
- The rate at which the surface area of the cube increases is 6 cm2/s.
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Community Answer
The volume of a cube is increasing at the rate of 18 cm3 per second. ...
Let, x be the length of an edge, V be the volume and S be the surface area of the cube.
S = 6x2
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The volume of a cube is increasing at the rate of 18 cm3 per second. When the edge of the cube is 12 cm, then the rate in cm2/s at which the surface area of the cube increases, isCorrect answer is '6'. Can you explain this answer?
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