Answer:
The largest number of vehicles he can put in one area is 28
There are 600 tenths of a second in one minute.
3/4 of a kilometer is 0.75 kilometer.
600 x 0.75 = 450
You would need 438.45 inches of wrapping paper to cover the triangular prism completely. But since you can’t really cut the paper to that exact decimal you would round to 439 inches. Don’t forget to cube your answer since it’s area. Final answer should look like 429in.2
Let <span>RR</span> be the radius of the sphere and let <span>hh</span> be the height of the cylinder centered on the center of the sphere. By the Pythagorean theorem, the radius of the cylinder is given by
<span><span><span>r2</span>=<span>R2</span>−<span><span>(<span>h2</span>)</span>2</span>.</span><span><span>r2</span>=<span>R2</span>−<span><span>(<span>h2</span>)</span>2</span>.</span></span>
The volume of the cylinder is hence
<span><span>V<span><span>=π<span>r2</span>h</span><span>=π<span>(<span>h<span>R2</span>−<span><span>h3</span>4</span></span>)</span>.</span></span></span><span><span>V<span>=π<span>r2</span>h</span></span><span>=π<span>(<span>h<span>R2</span>−<span><span>h3</span>4</span></span>)</span>.</span></span></span>
Differentiating with respect to <span>hh</span> and equating to <span>00</span> to find extrema gives
<span><span><span><span><span>dV</span><span>dh</span></span>=π<span>(<span><span>R2</span>−<span><span>3<span>h2</span></span>4</span></span>)</span>=0</span><span>∴<span>h0</span>=<span><span>2R</span><span>3‾√</span></span></span></span><span><span><span>dV</span><span>dh</span></span>=π<span>(<span><span>R2</span>−<span><span>3<span>h2</span></span>4</span></span>)</span>=0∴<span>h0</span>=<span><span>2R</span>3</span></span></span>
The second derivative of the volume with respect to <span>hh</span> is negative if <span><span>h>0</span><span>h>0</span></span> such that the volume is maximal at <span><span>h=<span>h0</span></span><span>h=<span>h0</span></span></span>. Substituting gives
<span><span><span>V<span>max</span></span>=<span><span>4π<span>R3</span></span><span>3<span>3‾√</span></span></span>.</span></span>
3.2 I think I'm not sure but I did try...