Chapter 2 - Maximum & Minimum Word Problems

Question 1

How do you know if there are asking you a minimum or maximum word problem? (2)

Answer 1

• When you’re trying to find the smallest/largest value of a quantity
• Based on a mathematical model (like volume, surface area, or area)

Question 2

What checklist should you use on every minimum/maximum word problem (8)

Answer 2

1. Define variables
   –> Use a let statement for x (dimension that changes)
   –> Assign a second variable (y) for second dimension
2. Write formulas (A, V, SA)
   –> Rectangle: A = xy
   –> Open Box w/ removed corner: V = x (L – 2x) (W – 2x), SA = LW − 4x²
   –> Square-Base Box: V = x²y, SA = 2x² + 4xy

–> Rectangle under a curve: A(x) = x · y
–> Cylinder: V = πr²h, SA = 2πr² + 2πrh
3. If you have 2 Variables & 2 Equations, use substitution
–> Pick one equation & solve one for a variable
–> Substitute expression into other formula for variable
–> Simplify until you have a single-variable function
4. Enter single-variable function into Y= and graph
–> Adjust window so min/max is visible
–> Use Table to Identify X & Y values
–> X min = 0 (dimensions cannot be negative)
–> X max = Value slightly larger than biggest x-value in table
–> Y min/Y max: extend beyond highest/lowest y-values
5. Use 2nd TRACE → min or max
6. Interpret X and Y of min or max
–> X-value = Input/Actual Variable of Equation
–> Y-value = Actual value that you’re solving for (A/V/SA/Cost)->

Question 3

What if there are two equations and two variables? (5)

Answer 3

1. Use substitution
2. Pick one equation & solve one for a variable
3. Substitute expression into other formula for variable
4. Simplify until you have a single-variable function
5. Plug that single-variable equation into y = in calculator

Question 4

How do you find the max or min using a graphing calculator? (8)

Answer 4

1. Enter single-variable function into Y= and graph
2. Adjust window so min/max is visible
3. Use Table to Identify X & Y values
4. X min: 0 (dimensions cannot be negative)
5. X max: Value slightly larger than x w/ biggest y in table
6. Y min/Y max: extend beyond highest/lowest y-values
7. Use 2nd TRACE to find minimum or maximum
8. Interpret X and Y of min or max

Question 5

What do the X and Y coordinates of the max/min mean? (2)

Answer 5

• X-value = Input/Actual Variable of Equation
• Y-value = Actual value that you’re solving for (A/V/SA/Cost)
  Ex:
  let x = length of cut
  Volume = x (20 - 2x) (18 - 2x)
  Maximum Coordinates (Length of Cut, Volume)

Question 6

What is the equation for the following:
1. Area of Rectangle
2. Volume of Open Box (cutting squares from corners) (2)
3. SA of Open Box (cutting squares from corners)
3. Volume of Square-Base Box (2)
4. Surface Area of Square-Base Box (2)
5. Rectangle Under a Curve (3)
6. Volume of Cylinder
7. Surface Area of Cylinder

Answer 6

1. Area of Rectangle: A = xy
2. Volume of an Open Box (cutting squares from corners):
   –> V = x (L - 2x) (W - 2x)
   –> x = size of square cut
3. Surface Area of an Open Box (cutting squares from corners):
   –> SA = LW - 4x²
4. Volume of a Square‑Base Box: V = x²y
   –> x = side of square base
   –> y = height
5. SA of a Square‑Base Box (open top):
   –> SA = x² + 4xy (NOT 2x² since top is removed)
   –> 1 Base & 4 Sides: x² + 4xy
6. Rectangle Under a Curve: x · f (x)
   –> x = Distance from origin to right side of rectangle
   –> width is 2x if it goes from -x to x
   –> Substitute y for f(x) of parabola (since y = f (x))
7. Volume of Cylinder (Base x Height): πr² · h
8. SA of Cylinder: SA = 2πr² + 2πrh

Question 7

How do you solve a word problem of a rectangle touching a function? (3)

Answer 7

1. A = xy –> A(x) = x · f(x)
   - Rewrites area as a single‑variable function
   - y is replaced w/ f(x) b/c top of rectangle touches curve
2. x= horizontal distance from ORIGIN to one side of rectangle
3. Width can be 2x if the rectangle goes from -x to x

Question 8

How do you model volume when cutting corners from a rectangle? (4)

Answer 8

1. Let x = side length of each square cut from the corners
2. Squares cut out –> sides folded up –> height of box is x
3. L & W each lose 2x (one cut on each side)
4. Volume: V(x) = x (L − 2x) (W − 2x)

Question 9

What do you do if the problem asks you to find surface area and the give you volume and vice versa? (3)

Answer 9

1. Use V & SA formulas together as a system of equations
2. Solve one equation for one variable & substitute into formula
3. Enter single‑variable equation into Y= on calculator

Question 10

What does it mean to maximize/minimize volume/surface area and how do you do it? (7)

Answer 10

1. Maximizing/Minimizing Volume: Make inside of the box as large/small as possible under material constraint (SA)
2. Maximizing/Minimizing Surface Area: Use the least or most material while keeping volume fixed
3. Use the constraint formula and solve for one variable
4. Substitute that expression into the other formula
5. Enter the resulting single‑variable equation into Y=
6. Use 2nd → TRACE → maximum or minimum
7. X = dimension; Y = smallest or largest possible SA or V

Question 11

What are the most common mistakes students make on max/min problems?

Answer 11

• Forgetting to subtract 2x from length/width when cutting corners
• Leaving two variables in the final equation
• Using negative dimensions
• Not adjusting the calculator window
• Misinterpreting the x-value (thinking it’s the answer when the question asks for something else)
• Forgetting that the max/min happens at the vertex of the curve

Question 12

After finding the coordinates of the minimum/maximum, how do you find the DIMENSIONS?

Answer 12

• Must find length, width and SOMETIMES height
• x‑value gives you key dimension (like base side or cut size).
• Plug that x‑value back into original equation to find missing dimension