Math  /  Geometry

Question14 Mark for Review As \begin{tabular}{|l|l|l|l|} \hline & & & \\ \hline & & & \\ & & & \\ & & & \\ \hlinexx & & & \\ \hlinexx & & & \\ \hlinexx \end{tabular}
The figure above represents a square sheet of cardboard with side length 20 inches. The sheet is cut and pieces are discarded. When the cardboard is folded, it becomes a rectangular box with a lid. The pattern for the rectangular box with a lid is shaded in the figure. Four squares with side length xx and two rectangular regions are discarded from the cardboard. Which of the following statements is true? (The volume VV of a rectangular box is given by V=lwhV=l w h. .)

Studdy Solution

STEP 1

1. The original cardboard is a square with side length 20 20 inches.
2. Four squares with side length x x are cut from the corners.
3. Two rectangular regions are also discarded.
4. The remaining cardboard forms a rectangular box with a lid.

STEP 2

1. Determine the dimensions of the rectangular box.
2. Calculate the volume of the box.
3. Analyze the given statements to determine which one is true.

STEP 3

Determine the dimensions of the rectangular box:
- After cutting out squares of side length x x from each corner, the length and width of the base of the box are reduced by 2x 2x each. - Therefore, the dimensions of the base of the box are (202x) (20 - 2x) by (202x) (20 - 2x) . - The height of the box is x x , as the squares cut out determine the height when folded.

STEP 4

Calculate the volume of the box using the formula V=l×w×h V = l \times w \times h :
V=(202x)×(202x)×x V = (20 - 2x) \times (20 - 2x) \times x

STEP 5

Analyze the given statements to determine which one is true:
- The volume of the box is expressed as V=(202x)2×x V = (20 - 2x)^2 \times x . - Evaluate each statement based on this expression to identify the true statement.
The true statement is the one that correctly describes the volume or dimensions of the box based on the expression derived.

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