Just had this one on an exam.
The surface area of a cube is increasing at a rate 4 meters squared/sec. How much is the volume of the cube increasing when the length of the cube is 10?
So a picture of a cube with a side labeld as 10.
Then figuring out what rates I have and what I needed, there's what I did. Am I right? I don't think so.
\(\displaystyle \
\L\begin{array}{l}
\frac{{dsa}}{{dt}} = 4m^2 /\sec \\
\frac{{dv}}{{dt}} = ??? \\
\frac{{da}}{{dt}} = ??? \\
\to sa = 6a \\
\to \frac{{dsa}}{{dt}} = 6\frac{{da}}{{dt}} \\
\to 4 = 6\frac{{da}}{{dt}} \\
\to \frac{{da}}{{dt}} = \frac{2}{3} \\
\to v = a^3 \\
\to \frac{{dv}}{{dt}} = 3a^2 \frac{{da}}{{dt}} \\
\to {\rm when a = }10x10x10 = 1000, \\
\to \frac{{dv}}{{dt}} = 3(1000)^2 \frac{2}{3} \\
\to \frac{{dv}}{{dt}} = 1000^2 (2) \\
\to \frac{{dv}}{{dt}} = 20,000,000!!! \\
\end{array}
\
\\)
Dear god I messed up on this question! How do you figure out related rates involving cubes? One of the only types of questions we did not go over before the exam. The only real probelm is I couldn't remember what the surface area or volume of a cube was (the formulas). I semi guessed the surface area was equal to 6 times the total area because a cube has 6 sides and that the volume of a cube is equale to the area cubed but then again, I don't think thats right. Then trying to figure out how to relate surface area to volume I could not remember.
The surface area of a cube is increasing at a rate 4 meters squared/sec. How much is the volume of the cube increasing when the length of the cube is 10?
So a picture of a cube with a side labeld as 10.
Then figuring out what rates I have and what I needed, there's what I did. Am I right? I don't think so.
\(\displaystyle \
\L\begin{array}{l}
\frac{{dsa}}{{dt}} = 4m^2 /\sec \\
\frac{{dv}}{{dt}} = ??? \\
\frac{{da}}{{dt}} = ??? \\
\to sa = 6a \\
\to \frac{{dsa}}{{dt}} = 6\frac{{da}}{{dt}} \\
\to 4 = 6\frac{{da}}{{dt}} \\
\to \frac{{da}}{{dt}} = \frac{2}{3} \\
\to v = a^3 \\
\to \frac{{dv}}{{dt}} = 3a^2 \frac{{da}}{{dt}} \\
\to {\rm when a = }10x10x10 = 1000, \\
\to \frac{{dv}}{{dt}} = 3(1000)^2 \frac{2}{3} \\
\to \frac{{dv}}{{dt}} = 1000^2 (2) \\
\to \frac{{dv}}{{dt}} = 20,000,000!!! \\
\end{array}
\
\\)
Dear god I messed up on this question! How do you figure out related rates involving cubes? One of the only types of questions we did not go over before the exam. The only real probelm is I couldn't remember what the surface area or volume of a cube was (the formulas). I semi guessed the surface area was equal to 6 times the total area because a cube has 6 sides and that the volume of a cube is equale to the area cubed but then again, I don't think thats right. Then trying to figure out how to relate surface area to volume I could not remember.