So here is the question I'm having a bit of trouble with!
"The Coast Guard is monitoring a giant iceberg in the approximate shape of a rectangular solid that is five times as long as the width across the front face. As the iceberg drifts south, the height above water is observed to decrease at the rate of two metres per week, and the width across the front is shrinking at three metres per week. Find the rate of loss of volume above the water when the height is 60m and the width of the face is 300m."
So here has been my approach towards this question, I end up getting an answer, but it's not correct 🙁
My approach:
I know that $l$=$5w$ and $\frac{dw}{dt} = \frac{-3m}{week}$ and $\frac{dh}{dt} = \frac{-2m}{week}$
So now I rearrange the volume formula for a rectangular prism, using $l=5w$ to $V=5w^2h$.
Now I take the new formula for the volume of a rectangular prism and differentiate it to get:
$\frac{dV}{dt} = 10w\frac{dw}{dt}h + 5w^2\frac{dh}{dt}$, then subbing in all the terms, $w=300m$ and $h=60m$ along with the other rates listed above, I get $\frac{dV}{dt} = 10(300)(-3)(60) + 5(300)^2(-2) = -1440000$.
But at the back of the book, it says its $-901800m^3/week$…so what am I doing wrong?
Please and thank you everyone!
Best Answer
I see two typos: the first parenthesis should be $300,$ not $600,$ and your result comes out (with that correction) to $-1,440,000.$ I agree with your answer. As a check, $300 \cdot 1500 \cdot 60-297 \cdot 1485 \cdot 58=1,413,000$ where I just applied the reductions.