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Wize University Calculus 1 Textbook > Applications of Differentiation

Optimization

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Optimization

The technique of finding a function's maximum and minimums can be applied to Optimization problems. These problems involve find the optimal value of a function, usually with a constraint equation.

Procedure for Optimization Problems

  1. Determine what you are minimizing/maximizing and draw a picture if possible
  2. Write an equation for the quantity you are trying to maximize or minimize (If there is more than 1 independent variable, create another equation relating the variables)
  3. Combine these to get an equation with an independent and a dependent variable
  4. Find and classify the critical points of this final combined equation
  5. Answer the question
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Example: Optimization

A farmer wishes to create a rectangular fenced enclosure for his animals. He has 600 ft. of fencing material and will construct the enclosure using a pre-existing fence as one side. Find the dimensions of the fenced area to maximize the enclosure.

1. Determine what you are minimizing/maximizing and draw a picture if possible
We want to MAXIMIZE the area of the rectangle

2. Write an equation for the quantity you are trying to maximize or minimize
*If there is more than 1 independent variable, create another equation relating the variables
The equation for area is A=l×wA=l\times w

We have 2 independent variables (length and width), so we need another equation that relates these 2 variables. Since the fence only has to make up 3 of the 4 sides of the rectangle, the equation is: 2l+w=6002l+w=600

3. Combine these to get an equation with an independent and a dependent variable
From the second equation, we have w=600−2lw=600-2l.
Substitute this into the area equation:
A=l(600−2l)A=l\left(600-2l\right)
A=600l−2l2A=600l-2l^2

4. Find and classify the critical points of this final combined equation
A′=600−4lA'=600-4l and A′′=−4A''=-4

To find the critical points, we set the first derivative = 0:
600−4l=0   →   l=150600-4l=0\ \ \ \to\ \ \ l=150

Check to make sure this is a maximum:
A′′(150)=−4<0A''\left(150\right)=-4<0
So, this critical point corresponds to a maximum.

5. Answer the question
Therefore, the dimensions that will maximize the enclosure are l=150 ftl=150\ ft and w=300 ftw=300\ ft.
The maximum area is 150×300=45000 ft2150\times300=45000\ ft^2.
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Example: Optimization

Find the point on the graph of the equation y=x,x≥0y=\sqrt{x}, x\geq 0, that is the closest to the point (3,0)(3,0).

1. Determine what you are minimizing/maximizing and draw a picture if possible

We are trying to minimize the distance between y=xy=\sqrt x and (3,0)(3,0).
2. Write an equation for the quantity you are trying to maximize or minimize
*If there is more than 1 independent variable, create another equation relating the variables

We want to optimize the distance formula d=(x−x0)2+(y−y0)2d = \sqrt{(x-x_0)^2 + (y-y_0)^2 }

However, for purposes of the derivative, it is equivalent and easier to use the square of the distance formula.

d2=(x−x0)2+(y−y0)2d^2 = (x-x_0)^2 + (y-y_0)^2

Since this equation has more than 1 independent variable, substitute the constraint equation y=xy=\sqrt x

3. Combine these to get an equation with an independent and a dependent variable

d2=(x−x0)2+(x−y0)2d^2 = (x-x_0)^2 + (\sqrt x-y_0)^2

Also we may substitute (3,0)(3,0) for x0x_0 and y0y_0

d2=(x−3)2+(x−0)2d^2 = (x-3)^2 + (\sqrt x-0)^2

=(x−3)2+x=(x-3)^2+x

4. Find and classify the critical points of this final combined equation

ddx[d2]=ddx(x−3)2+x\displaystyle \frac{d}{dx}[d^2]=\frac{d}{dx}(x-3)^2+x

⇒0=2(x−3)+1\Rightarrow0=2(x-3)+1

⇒x−3=−12\displaystyle\Rightarrow x-3=-\frac{1}{2}

⇒x=52→critical point\displaystyle \Rightarrow x=\frac{5}{2} \rightarrow \text{critical point}

f′′(x)=2>0∴ local, min, at x=52\displaystyle f''(x) = 2 > 0 \therefore \text{ local, min, at } x= \frac{5}{2}

5. Answer the question

Substitute x=52\displaystyle x=\frac{5}{2} into y=x  ⟹  y=52\displaystyle y=\sqrt x \implies y=\sqrt{\frac{5}{2}}

So the point (52,52)\left(\dfrac{5}{2},\sqrt{\dfrac{5}{2}}\right) on x\sqrt{x} is closest to (3,0)(3,0)

Four square corners are cut from a rectangular sheet measuring 160 cm by 160cm, and the remaining shape is bent to form an open-topped box. Find the side length of the cut-out square that maximizes the volume of the box.

Enter your answer in cm, do NOT include units


Find the area of the largest rectangle that has two vertices on the xx-axis and two vertices lying on the graph of y=8−x2y=8-x^2 with
−8≤x≤8-\sqrt{8}\leq x\leq \sqrt{8}.



Extra Practice
Optimization
A metal sheet is cut from the shaded area and bend to form a box.
Find the value of xx which maximizes the volume of the box.
Optimization
A cylindrical can without a top is made to contain 81π81\pi cm3cm^3 of liquid. Determine the dimensions (radius and height) of the can that minimize the area of metal used. Don't forget to confirm that your answer is a minimum!
Optimization
Suppose you have a rectangle of perimeter 20. Find the dimensions that will maximize the area.
Optimization
Suppose you have a rectangle of perimeter 20. Find the dimensions that will maximize the area.
Optimization: Rectangular Field
We need to enclose a field with a rectangular fence. We have 500 ft of fencing material and a building is on one side of the field and so won’t need any fencing there. Determine the dimensions of the field that will enclose the largest area, and the maximum area.
Practice: Optimization
A metal cylinder without a top is made to contain 64π64\pi cm3 of glue. Determine the dimensions of the can that will minimize the amount of metal used.
Optimization: Cylindrical Can
A cylindrical can without a top is made to contain 81π81\pi cm3 of liquid. Determine the dimensions of the can that minimizes the amount of material used.
Optimization
Find the dimensions of the rectangle of largest area that has its base on the xx-axis, its other two vertices above the xx-axis and is laying on the parabola y=20−x2y=20-x^2.
Optimization
A metal cylinder without a top is made to contain 64π64\pi cm3 of glue. Determine the dimensions of the can that will minimize the amount of metal used.
Optimization
Find the minimum vertical distance between the graphs of y1=x2+x+10y_1=x^2+x+10 and y2=−x2+x+2y_2=-x^2+x+2.
Optimization: Volume
Four square corners are cut from a rectangular sheet measuring 160 cm by 160cm, and the remaining shape is bent to form an open-topped box. Find the side length of the cut-out square that maximizes the volume of the box.
Optimization: Rectangular Box
A square of size xx is cut out of the corners of a metal sheet of size 60×\times40 units, then the sheet is bent to form a box. Find the value of xx which maximizes the volume of the box.
Optimization: Cylindrical Can
Q.\textbf{Q.} A new competitor on the energy-drink market wants to make a cylindrical can that will hold 1.5 liters of liquid. Determine the dimensions of the can that will minimize the amount of material used in its construction. Give a qualitative reasoning only as to why this is a minimum. Do not simplify.
Optimization: Rectangular Field
We need to enclose a field with a rectangular fence. We have 500 ft of fencing material and a building is on one side of the field and so won’t need any fencing there. Determine the dimensions of the field that will enclose the largest area, and the maximum area.
Practice: Time of Travel
Q.\textbf{Q.} ABC is a right triangle, with the right angle at point B. Sides AB and BC have length L. We want to go from B to A by first going to an intermediate point D on BC with speed v1v_1 and then going from D to A with speed v2v_2. Find the position of D which minimizes the travel time, given that v1=3v2v_1=3v_2.
Practice: Triangle Inside Circle
Q.\textbf{Q.} Find the coordinates of the point PP which lies on the curve 4−x2\sqrt{4-x^2} for which the area of the right triangle formed by the x-axis, the point PP and a point of intersection of the curve with the x-axis is maximum.
Optimization: Rectangle Inside Parabola
Find the dimensions of the rectangle of largest area that has its base on the xx-axis, and its other two vertices above the x-axis on the parabola y=20−x2y=20-x^2.
Optimization: Cylindrical Can
A cylindrical can without a top is made to contain 81π81\pi cm3 of liquid. Determine the dimensions of the can that minimizes the amount of material used.
Optimization: Square and Circle
Four feet of wire is to be used to form a square and a circle. How much wire should be used for the square and how much for the circle in order to enclose the least, respectively the greatest, total area?
Optimization
Consider a cone with a height 2 metres and whose circular base has a radius 1 metre. Find the dimensions of the circular cylinder of largest volume that can be contained in the cone. (The base of the cylinder lies at the base of the cone.)
Optimization
Josh wants to build a rectangular fence around his plot of land, where one of the sides will face the street. The street facing side of the fence costs $8/foot and the other 3 sides of the fence cost $2/foot. What are the dimensions of this fenced in area that will minimize his cost if he wants an area of 1000 ft2?
The sum of two positive integers is 20. What is the smallest possible value for their squares?
A metal sheet is cut from the shaded area and bend to form a box.
Find the value of xx which maximizes the volume of the box.
A metal sheet is cut from the shaded area and bend to form a box.
Find the value of xx which maximizes the volume of the box.
Optimization
Find the area of the largest rectangle that has two vertices on the xx-axis and two vertices lying on the graph of y=8−x2y=8-x^2 with
−8≤x≤8-\sqrt{8}\leq x\leq \sqrt{8}.
Find the area of the largest rectangle that has two vertices on the xx-axis and two vertices lying on the graph of y=8−x2y=8-x^2 with −8≤x≤8-\sqrt{8}\leq x\leq \sqrt{8}.
Practice: Word Problem Max/Min (Similar to April 2018 Q45)
Find two integers that have a product of 16 and a minimum sum.
Practice: Optimization with Graph
Practice Question: Optimization
Find the dimensions of the rectangle of largest area that has its base on the xx-axis, its other two vertices above the xx-axis and is laying on the parabola y=20−x2y=20-x^2.
Practice: Optimization
Practice Question: Optimization
A metal cylinder without a top is made to contain 64π64\pi cm3 of glue. Determine the dimensions of the can that will minimize the amount of metal used.
Practice: Optimization
Practice Question: Optimization
Four square corners are cut from a rectangular sheet measuring 160 cm by 160cm, and the remaining shape is bent to form an open-topped box. Find the side length of the cut-out square that maximizes the volume of the box.
Enter your answer in cm, do NOT include units
Optimization
Find the point on y=x2y=x^2 closest to the point (0,1)(0,1).
Optimization: Velocity of a wave
The velocity of a wave of length LL in deep water is
v=KLC+CLv=K\sqrt{\frac{L}{C}+\frac{C}{L}}
where K and C are known positive constants. What is the length of the wave that gives the minimum velocity?
Optimization
Find the point or points on the parabola y=−2x2+1y=-2x^2+1 closest to the origin.
Optimization
Find two positive integers such that the sum of the first number and four times the second number is 1000 and the product the numbers is as large as possible.
The first number:
500
The second number:
125
A rectangle has two points on the x-axis and two other points on the parabola with the equation y=x2−4.y=x^2-4. If it has the maximum area, find its dimensions?
Optimization
A farmer wishes to create a rectangular fenced enclosure for his animals. He has 600 m of fencing material and will construct the enclosure using a pre-existing fence as one side. Find the dimensions of the fenced area to maximize the enclosure.
Optimization
What angle θ\theta between two edges of length 3 will result in an isosceles triangle with the largest area?
Optimization
We want to construct a box whose base length is 3 times the base width. The material used to build the top and bottom cost $10/ft2\$ 10/ft^2 and the material used to build the sides cost $6/ft2\$6/ft^2. If the box must have a volume of 60 ft3ft^3 determine the dimensions that will minimize the cost to build the box.
Optimization
Suppose you have a rectangle of perimeter 20. Find the dimensions that will maximize the area.
Suppose you have a rectangle with area 20 m2. Find the dimensions that will maximize the perimeter.
A metal sheet is cut at the corners (the shaded areas) and bent to form a box. Find the value of xx which maximizes the volume of the box.
Find the coordinates of the point PPwhich lies on the curve
y=4−x2y=\sqrt{4-x^2} and for which the area of the right triangle ABP is maximum.
Find the area of the largest rectangle that has two vertices on the xx-axis and two vertices lying on the graph of y=8−x2y=8-x^2 with
−8≤x≤8-\sqrt{8}\leq x\leq \sqrt{8}.
Practice: Triangle Inside Circle
Q.\textbf{Q.} Find the coordinates of the point PP which lies on the curve 4−x2\sqrt{4-x^2} for which the area of the right triangle formed by the x-axis, the point PP and a point of intersection of the curve with the x-axis is maximum.
Find the points on the graph of the curve xy2=54xy^2=54 that are closest to the origin.
Practice: Time of Travel
Q.\textbf{Q.} ABC is a right triangle, with the right angle at point B. Sides AB and BC have length L. We want to go from B to A by first going to an intermediate point D on BC with speed v1v_1 and then going from D to A with speed v2v_2. Find the position of D which minimizes the travel time, given that v1=3v2v_1=3v_2.
Optimization: Cylindrical Can
A cylindrical can without a top is made to contain 81π81\pi cm3 of liquid. Determine the dimensions of the can that minimizes the amount of material used.
Optimization: Rectangular Box
A square of size xx is cut out of the corners of a metal sheet of size 60×\times40 units, then the sheet is bent to form a box. Find the value of xx which maximizes the volume of the box.
Optimization: Rectangle Inside Parabola
Find the dimensions of the rectangle of largest area that has its base on the xx-axis, and its other two vertices above the x-axis on the parabola y=20−x2y=20-x^2.
Optimization: Square and Circle
Four feet of wire is to be used to form a square and a circle. How much wire should be used for the square and how much for the circle in order to enclose the least, respectively the greatest, total area?
Optimization: Rectangular Field
We need to enclose a field with a rectangular fence. We have 500 ft of fencing material and a building is on one side of the field and so won’t need any fencing there. Determine the dimensions of the field that will enclose the largest area, and the maximum area.
Optimization: Cylindrical Can
Q.\textbf{Q.} A new competitor on the energy-drink market wants to make a cylindrical can that will hold 1.5 liters of liquid. Determine the dimensions of the can that will minimize the amount of material used in its construction. Give a qualitative reasoning only as to why this is a minimum. Do not simplify.
Optimization
A rectangle has two points on the x-axis and two other points on the parabola with the equation y=x2−4.y=x^2-4. If it has the maximum area, find its dimensions?
A box-shaped shipping crate with a square base is designed to have a volume of 16 m3. The material used to make the base costs twice as much per square meter as the material in the sides, and the material used to make the top costs half as much per square meter as the material in the sides. Suppose the material to makes the sides costs $200 per square meter. What are the dimensions of the crate that minimize the cost of the materials?
Optimization with minimizing surface area
A cylindrical can without a top is made to contain 64π64\picm3 of liquid. Determine the dimensions of the can that minimize the area of metal used.
Optimization
Find the dimensions of the rectangle of largest area that has its base on the x-axis, its other two vertices above the x-axis and is laying on the parabola y = 20 - x2.
Optimization
Find the maximum area of a rectangle bound by the parabola y=4−x2y=4-x^2 and the xx-axis.
Determine the dimensions of a cylindrical container that minimizes the amount of material used yet gives a volume of 1.5 liters.
Optimization
A squares of length of xx are cut from the corners of a 4040 by 6060 sheet of metal. The sides are then bent up to form a box. Find the value of xx which maximizes the volume of the box.
Find the coordinates of the point PPwhich lies on the curve
y=4−x2y=\sqrt{4-x^2} and for which the area of the right triangle ABP is maximum.
Find the area of the largest rectangle that has two vertices on the xx-axis and two vertices lying on the graph of y=8−x2y=8-x^2 with −8≤x≤8-\sqrt{8}\leq x\leq \sqrt{8}.
Optimization
Determine the dimensions of a cylindrical container that minimizes the amount of material used to give a volume of 1.5 liters.
Optimization
A metal sheet is cut from the shaded area and bend to form a box.
Find the value of xx which maximizes the volume of the box.
Find the dimensions of a right-angle triangle with hypotenuse 10cm that will maximize its area.
Optimization
An open-top square-based box is to be made using cardboard of different thickness. The base (bottom) of the box uses 3 layers of cardboard, and the sides of the box use 2 layers of cardboard. If the volume of the box must be 6000m3, find the dimensions of the box that will minimize the material used.
Optimization
Find the minimum vertical distance between the graphs of y1=x2+x+10y_1=x^2+x+10 and y2=−x2+x+2y_2=-x^2+x+2.
Optimization
Find the dimensions of the rectangle of largest area that has its base on the xx-axis, its other two vertices above the xx-axis and is laying on the parabola y=20−x2y=20-x^2.
Optimization
A metal cylinder without a top is made to contain 64π64\pi cm3 of glue. Determine the dimensions of the can that will minimize the amount of metal used.
Optimization: Volume
Four square corners are cut from a rectangular sheet measuring 160 cm by 160cm, and the remaining shape is bent to form an open-topped box. Find the side length of the cut-out square that maximizes the volume of the box.
A small indie company needs to sell 10000 copies of their latest video game in order to keep their sponsors. Each copy of the game is sold for $50. However, in order to produce all these copies, they need to hire more people. The workers get paid a salary of $1000 each and each worker can pump out a game in 2 hours. Additionally, a supervisor has to be hired who receives $3000 upfront, and gets paid $20 per hour. Minimize the costs needed to produce all 10,000 copies and determine how much money they would make.
The BoxStore sells a special kind of box for companies to store their products. The special part about the box is that the bottom must be a square and that it has no top part (it has four sides and a bottom). Typically, a box is 1  m31 \;m^3 in volume. However, due to changes in policy, BoxStore can now only use 10000  cm210000 \; cm^2 of material to create their box.
A small indie company needs to sell 10000 copies of their latest video game in order to keep their sponsors. Each copy of the game is sold for $50. However, in order to produce all these copies, they need to hire more people. The workers get paid a salary of $1000 each and each worker can pump out a game in 2 hours. Additionally, a supervisor has to be hired who receives $3000 upfront, and gets paid $20 per hour. Minimize the costs needed to produce all 10,000 copies and determine how much money they would make.
Optimization
IWear needs to hire workers to produce 2250 glasses for their latest product at a factory. One worker can produce 5 glasses per hour, and is paid $10 for each pair of glasses they produce. In addition, the factory supervisor must be paid $30 per hour while the workers create the new product. Additionally, each worker is paid $15 up front for being hired (but not the supervisor). How many workers should IWear hire in order to minimize the cost of making these glasses?
Optimization
The BoxStore sells a special kind of box for companies to store their products. The special part about the box is that the bottom must be a square and that it has no top part (it has four sides and a bottom). Typically, a box is 1  m31 \;m^3 in volume. However, due to changes in policy, BoxStore can now only use 10000  cm210000 \; cm^2 of material to create their box. What is the largest box they can now make and is it smaller or larger than the original?
Find the dimensions of the rectangle of largest area that has its base on the xx-axis and other two vertices above the xx-axis and laying on the parabola y=20−x2y=20-x^2.
Optimization
A cylindrical cone without a top is made to contain 81π81\pi cm3cm^3 of liquid. Determine the dimensions of the can that minimize the area
of metal used.
Optimization
ABC is a right triangle, with right angle at point BB. Sides
AB and BC have length L. We want to go from C to A by first going to an intermediate point D on BC with velocity v1v_1 and then going away from D to A with velocity v2v_2. Find the position of D which minimizes the travel time given that v1=3v2v_1=3v_2.
Optimization
A poster has a total area of 200 cm2cm^2with 1 cmcmmargins at the
bottom and sides and 2 cmcm at the top. What dimensions gives the largest printed area?
A metal sheet is cut from the shaded area and bend to form a box.
Find the value of xx which maximizes the volume of the box.