Math 220 March 7 I. If 1200 cm2 of material is available to make a ...
Math 220 March 7 I. If 1200 cm2 of material is available to make a ...
Math 220 March 7 I. If 1200 cm2 of material is available to make a ...
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I. <strong>If</strong> <strong>1200</strong> cm 2 <strong>of</strong> <strong>material</strong> <strong>is</strong> <strong>available</strong> <strong>to</strong> <strong>make</strong> a box with a square base<br />
and an open <strong>to</strong>p, find the largest possible volume <strong>of</strong> the box.<br />
Answer:<br />
V olume = x 2 y<br />
SurfaceArea = <strong>1200</strong> = 4x 2 + xy =⇒ y =<br />
v = x 2 y<br />
v(x) = x 2 <strong>1200</strong> − 4x2<br />
( )<br />
x<br />
v(x) = x(<strong>1200</strong> − 4x 2 )<br />
v ′ (x) = (<strong>1200</strong> − 4x 2 ) + x(−8x)<br />
0 = <strong>1200</strong> − 12x 2<br />
0 = 100 − x 2<br />
x = ±10<br />
v ′′ (x) = −24x<br />
v ′′ (10) < 0<br />
<strong>1200</strong> − 4x2<br />
x<br />
There <strong>is</strong> a maximum when x = 10 and y = 12<br />
The largest possible volume <strong>of</strong> the box <strong>is</strong> <strong>1200</strong> cm 3 .<br />
II. Find the point on the curve y = √ x that <strong>is</strong> closest <strong>to</strong> point (4,0).<br />
Answer:<br />
D<strong>is</strong>tance = (x − 4) 2 + (y − 0) 2<br />
3
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