Questions: Question 3 1 pts Note: This question closely resembles one of the examples in the supplemental video linked below. Sam is making an open box (no top) from a tin sheet which is 8 inches by 3 inches by cutting out identical squares from each corner and bending up the resulting flaps. (a) Which of the following expresses the volume of the box as a function of x ? (Type A, B or C.) A. V=x * 8 * 3 B. V=x *(8-x) *(3-x) C. V=x *(8-2x) *(3-2x) (b) Find the width of the cut (i.e. the value of x ) that would result in the largest box. x= inches (Enter your answer as an exact fraction or round your answer to two decimal places.)

Question 3 1 pts Note: This question closely resembles one of the examples in the supplemental video linked below. Sam is making an open box (no top) from a tin sheet which is 8 inches by 3 inches by cutting out identical squares from each corner and bending up the resulting flaps. (a) Which of the following expresses the volume of the box as a function of x ? (Type A, B or C.) A. V=x * 8 * 3 B. V=x *(8-x) *(3-x) C. V=x *(8-2x) *(3-2x) (b) Find the width of the cut (i.e. the value of x ) that would result in the largest box. x= inches (Enter your answer as an exact fraction or round your answer to two decimal places.)
Transcript text: Question 3 1 pts Note: This question closely resembles one of the examples in the supplemental video linked below. Sam is making an open box (no top) from a tin sheet which is 8 inches by 3 inches by cutting out identical squares from each corner and bending up the resulting flaps. (a) Which of the following expresses the volume of the box as a function of $x$ ? $\square$ (Type A, B or C.) A. $V=x \cdot 8 \cdot 3$ B. $V=x \cdot(8-x) \cdot(3-x)$ C. $V=x \cdot(8-2 x) \cdot(3-2 x)$ (b) Find the width of the cut (i.e. the value of $x$ ) that would result in the largest box. $x=$ $\square$ inches (Enter your answer as an exact fraction or round your answer to two decimal places.)
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Solution

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Solution Steps

Step 1: Define the dimensions of the box
  • The original sheet is 8 inches by 3 inches.
  • Squares of side length \( x \) are cut from each corner.
Step 2: Determine the new dimensions after cutting
  • After cutting out squares of side length \( x \), the new dimensions of the box will be:
    • Length: \( 8 - 2x \)
    • Width: \( 3 - 2x \)
    • Height: \( x \)
Step 3: Express the volume of the box as a function of \( x \)
  • The volume \( V \) of the box is given by the product of its length, width, and height: \[ V = (8 - 2x)(3 - 2x)x \]
  • Simplify the expression: \[ V = x(8 - 2x)(3 - 2x) \] \[ V = x(24 - 16x - 6x + 4x^2) \] \[ V = x(4x^2 - 22x + 24) \] \[ V = 4x^3 - 22x^2 + 24x \]

Final Answer

  • The volume of the box as a function of \( x \) is: \[ V = 4x^3 - 22x^2 + 24x \]
  • This corresponds to option C.
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