Lecture Twenty-One Practice

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\newcommand {\ansPEightAOne }[0]{2}$

Practice problems for Lecture Twenty One Content

1 : A Farmer has $2400$ feet of fencing and wants to fence off a rectangular field that boarders a straight river. He needs no fence along the river. What are the dimensions of the field that has the largest area? (List the smallest dimension first)

$\answer {\ansPOneAOne }$ feet $\times$ $\answer {\ansPOneATwo }$ feet

2 : Find the point on the parabola $y^2 = 2x$ that is closest to the point $(1,4)$ . $(\answer {\ansPTwoAOne }, \answer {\ansPTwoATwo })$

3 : Find the area of the largest rectangle that can be inscribed in a semicircle of radius 5. $\answer {\ansPThreeAOne }$ square units.

4 : Find the positive numbers whose product is $100$ and whose sum is the smallest possible. (List the smallest number first).

$\answer {\ansPFourAOne }, \answer {\ansPFourATwo }$ .

5 : If $1200$ square centimeters of material is available to make a box with a square base and an open top, find the largest possible volume of the box. $\answer {\ansPFiveAOne }$ cubic centimeters.

6 : Find the largest area of an isosceles triangle inscribed in a circle of radius 3. $\answer {\ansPSixAOne }$ .

7 : The top and bottom margins of a poster $6$ cm each, and the side margins are $4$ cm each. If the area of the printed material on the poster is fixed at $384$ square centimeters, find the dimensions of the poster of smallest area. (list the smallest dimension first)

$\answer {\ansPSevenAOne }$ cm $\times$ $\answer {\ansPSevenATwo }$ cm

8 : A box with an open top is to be constructed from a square piece of cardboard that is three feet on each side, by cutting out a square from each of the four corners and bending up the sides. What is the largest volume that such a box can have? $\answer {\ansPEightAOne }$ cubic feet.

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