1. A company wants to fence a rectangular storage area consisting of three separ
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Question
1. A company wants to fence a rectangular storage area consisting of three separate compartments of equal size as shown in the figure below. The fencing is shown in blue. Suppose that the total length of fencing is 300 feet. What is the largest possible area you can enclose with this structure? Give an exact answer with correct units.
2. A company wants to fence a rectangular storage area consisting of three separate compartments of equal size as shown in the figure below. The fencing is shown in blue. Suppose that the total enclosed floor area is 2000 square feet. What dimensions would use the least amount of fence? Be accurate to one decimal place and include units. L= W =
3. A trapezoid is inscribed in the first quadrant under a parabola as shown below.
a) Write a formula for the area, A, of the trapezoid. Use both x and y. Do not use any decimals.A(x,y) = Incorrect: Your answer is incorrect.
b) Rewrite your formula so that area is a function of x only. A(x) =
c) Find the maximum possible area. Be accurate to 3 decimal places. No units. Amax =
4. A rectangle is inscribed under a parabola as shown below.
a) Write a formula for the area, A, of the rectangle. Use both b and y. A(b,y) =
b) Rewrite your formula so that area is a function of x. You will have to figure out what to substitute for b and y. A(x) =
c) Find the maximum possible area. Be accurate to 3 decimal places. No units. Amax =
5. A triangle is inscribed under a parabola as shown below.
a) Write a formula for the area, A, of the triangle. Use both b and y. A(b,y) =
b) Rewrite your formula so that area is a function of x. A(x)=
c) Find the maximum possible area. Be accurate to 3 decimal places. No units. Amax =
6. A corral is to be built by joining three sides of a rectangle to a semicircle as shown figure below. The diameter of the semicircle is the same as one side of the rectangle.
a) Write a formula for the total length of fence used in this construction. Your formula must use both x and y. Length =
b) Write a formula for the total area enclosed in the corral. Use x and y. Area =
7. A corral is to be built by joining three sides of a rectangle to a semicircle as shown figure below. The diameter of the semicircle is the same as one side of the rectangle.
The total amount of fencing available is 500 feet. What is the largest possible area you can enclose? Be accurate to the nearest whole number. Include correct units.
8. A box has two square sides and one open rectangular face, as shown in the figure below.
Suppose that the total surface area (five sides) is 600 square centimeters. What dimensions will give the largest volume? Be accurate to one decimal place and include units. x= y=
9. The price of a commodity is a function of time,p(t),with p in dollars/lb and t in months. In the domain0 ? t ? 2months, the rate of change of price is dp/dt= 11.5 sin(2.8t +0.2) Answer the following questions.
Be accurate to 4 decimal places and use correct units.
1. Locate the instant in time (in the given domain) when the price is greatest. t=
2. Locate the instant in time (in the given domain) when the price is going up at its fastest rate. t=
3. At that instant, how fast is the price changing?
4. Locate the instant in time (in the given domain) when the price is going down as fast as possible. t=
5. At that instant, how fast is the price changing?
1. A company wants to fence a rectangular storage area consisting of three separate compartments of equal size as shown in the figure below. The fencing is shown in blue. Suppose that the total length of fencing is 300 feet. What is the largest possible area you can enclose with this structure? Give an exact answer with correct units. 2. A company wants to fence a rectangular storage area consisting of three separate compartments of equal size as shown in the figure below. The fencing is shown in blue. Suppose that the total enclosed floor area is 2000 square feet. What dimensions would use the least amount of fence? Be accurate to one decimal place and include units. L= W = 3. A trapezoid is inscribed in the first quadrant under a parabola as shown below. a) Write a formula for the area, A, of the trapezoid. Use both x and y. Do not use any decimals.A(x,y) = Incorrect: Your answer is incorrect. b) Rewrite your formula so that area is a function of x only. A(x) = c) Find the maximum possible area. Be accurate to 3 decimal places. No units. Amax = 4. A rectangle is inscribed under a parabola as shown below. a) Write a formula for the area, A, of the rectangle. Use both b and y. A(b,y) = b) Rewrite your formula so that area is a function of x. You will have to figure out what to substitute for b and y. A(x) = c) Find the maximum possible area. Be accurate to 3 decimal places. No units. Amax = 5. A triangle is inscribed under a parabola as shown below. a) Write a formula for the area, A, of the triangle. Use both b and y. A(b,y) = b) Rewrite your formula so that area is a function of x. A(x)= c) Find the maximum possible area. Be accurate to 3 decimal places. No units. Amax = 6. A corral is to be built by joining three sides of a rectangle to a semicircle as shown figure below. The diameter of the semicircle is the same as one side of the rectangle. a) Write a formula for the total length of fence used in this construction. Your formula must use both x and y. Length = b) Write a formula for the total area enclosed in the corral. Use x and y. Area = 7. A corral is to be built by joining three sides of a rectangle to a semicircle as shown figure below. The diameter of the semicircle is the same as one side of the rectangle. The total amount of fencing available is 500 feet. What is the largest possible area you can enclose? Be accurate to the nearest whole number. Include correct units. 8. A box has two square sides and one open rectangular face, as shown in the figure below. Suppose that the total surface area (five sides) is 600 square centimeters. What dimensions will give the largest volume? Be accurate to one decimal place and include units. x= y= 9. The price of a commodity is a function of time,p(t),with p in dollars/lb and t in months. In the domain0 ? t ? 2months, the rate of change of price is dp/dt= 11.5 sin(2.8t +0.2) Answer the following questions. Be accurate to 4 decimal places and use correct units. 1. Locate the instant in time (in the given domain) when the price is greatest. t= 2. Locate the instant in time (in the given domain) when the price is going up at its fastest rate. t= 3. At that instant, how fast is the price changing? 4. Locate the instant in time (in the given domain) when the price is going down as fast as possible. t= 5. At that instant, how fast is the price changing?Explanation / Answer
1.)
Fence = 4W + 2L = 300
Area = WL
or Area = (150-2W)W
Max Area = differentiate with repect to W
150 - 4W = 0
W = 37.5 feet
L = 75 feet
Therefore Max Area = 2812.5 square feet.
2.)
Area = WL = 2000
Fence = 4W + 2L
or Fence = 4(2000/L) + 2L
Min Fence = differentiate with repect to L
-8000/L^2 + 2 = 0
L^2 = 4000
L = 63.24 feet
Therefore
W = 31.62 feet
3.)
a)Area of trapezium = 1/2(sum of parallell sides)*perpendicular
= 1/2(1+y)x
b)Area of trapezium = 1/2(sum of parallell sides)*perpendicular
= 1/2(2-x^2)x
c)Max area = d/dx(Area) = 0
= x = (2/3)^.5
therefore area = 2/3(2/3)^.5
= .544 units
4.)
a)Area of rectangle = length * breadth
= b * y
b)Area of rectangle = b * y
= 2x * (1-x^2)
c)Max area = d/dx(Area) = 0
= x = (1/3)^.5
therefore area = 4/3(1/3)^.5
= .769 units
5.)
a)A = 1/2*b*y
b)A(x) = 1/2*(2x)*(1-x^2)
c)Max Area = d/dx(Area) = 0
= x = (1/3)^.5
therefore area = 2/3(1/3)^.5
= .384 units
6.)
a)Length = x + 2y + pi*x/2
b)Area = xy + (pi*x^2)/8
7)x + 2y + pi*x/2 = 500
there fore y= (500 - x - pi*x/2)/2
Max Area is d/dx of area =0
d/dx[x(500 - x - pi*x/2)/2 + + (pi*x^2)/8] = 0
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