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>
Answer:
Its 4.3
Step-by-step explanation:
If you graph it, y is the height an t is the time. At a height of 0 it had been in the air for 4.3 seconds
Answer:
g(f(-1)) = 48
Step-by-step explanation:
given the following
f(x) = 5x² + 2
g(x) = x²-1
g(f(x)) = g(5x²+2)
Replace x with 5x² + 2 in g(x) as shown;
g(5x²+2) = (5x²+2))²- 1
g(f(x)) = (5x²+2))²- 1
g(f(-1)) = (5(-1)²+2))²- 1
g(f(-1)) = (5+2))²- 1
g(f(-1)) = (7)²- 1
g(f(-1)) = 49 - 1
g(f(-1)) = 48
It's the first choice. 9b = 11.07.
Answer:
Yes
Step-by-step explanation:
Each value only maps out to one value, so that means it is a function.