1.) Find the dimensions of a rectangle with area 1,000 m 2 whose perimeter is as
ID: 2842074 • Letter: 1
Question
1.) Find the dimensions of a rectangle with area 1,000 m2 whose perimeter is as small as possible.
m (smaller value) m (larger value)2.)Consider the following problem: A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have. (a) Draw several diagrams to illustrate the situation, some short boxes with large bases and some tall boxes with small bases. Find the volumes of several such boxes.
(b) Draw a diagram illustrating the general situation. Let x denote the length of the side of the square being cut out. Let y denote the length of the base.
(c) Write an expression for the volume V in terms of x and y.
V =
(d) Use the given information to write an equation that relates the variables x and y.
(e) Use part (d) to write the volume as a function of x.
V(x) =
(f) Finish solving the problem by finding the largest volume that such a box can have.
V = ft3
3)A rectangular storage container with an open top is to have a volume of 10 m3. The length of this base is twice the width. Material for the base costs $10 per square meter. Material for the sides costs $6 per square meter. Find the cost of materials for the cheapest such container. (Round your answer to the nearest cent.)
$
4)A right circular cylinder is inscribed in a cone with height h and base radius r. Find the largest possible volume of such a cylinder.
5)A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle. See the figure below.) If the perimeter of the window is 8 ft, find the value of x so that the greatest possible amount of light is admitted.
x = ft
6)A woman at a point A on the shore of a circular lake with radius 2 mi wants to arrive at the point C diametrically opposite A on the other side of the lake in the shortest possible time (see the figure). She can walk at the rate of 4 mi/h and row a boat at 2 mi/h. For what value of the angle ? shown in the figure will she minimize her travel time?
? =
7)A poster is to have an area of 240 in2 with 1 inch margins at the bottom and sides and a 2 inch margin at the top. Find the exact dimensions that will give the largest printed area. width in height in
Explanation / Answer
Inasmuch as a square is also a type of rectangle, then.
= ?1,000
= 31.622777
Answer: 31.622777 meters by 31.622777 meters
Perimeter (the smallest as can be):
= 31.622777 * 4
= 126.4911 meters
c) V=y^2 x
d) y+2x=3
e) V(x) = (3-2x)(3-2x)x
f)
V(x) = (9x-12x^2+4x^3)
V'(x) = 9-24x+12x^2 = 0
12x^2-6x-18x+9=0
6x(2x-1)-9(2x-1)=0
(2x-1)(6x-9)=0
x=1/2 or x=9/6=3/2
V''(x) = 24x-24
At x=1/2 , V''(x) < 0, volume is maximized
Volume = (3-1)(3-1)(1/2) = 2 ft^3
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