A student wonders if she triples each measurement of a cereal box, will she need three times as much cardboard to make the new box. How would you help her decide?
A student wonders if she triples each measurement of a cereal box, will she need three times as much cardboard to make the new box. How would you help her decide?
To help the student decide whether she'll need three times as much cardboard to make the new box, we can break down the problem and do some calculations.
Let's assume the original measurements of the cereal box are:
- Length: L
- Width: W
- Height: H
To triple each measurement, we will multiply each dimension by 3:
- New Length: 3L
- New Width: 3W
- New Height: 3H
Now, to determine the amount of cardboard required for the new box, we need to calculate the surface area of both the original and new boxes. The surface area of a rectangular box can be found using the formula:
Surface Area = 2 * (Length * Width + Width * Height + Height * Length)
For the original box, the surface area would be:
Original Surface Area = 2 * (L * W + W * H + H * L)
For the new, tripled box, the surface area would be:
New Surface Area = 2 * (3L * 3W + 3W * 3H + 3H * 3L)
Simplifying the expressions, we get:
Original Surface Area = 2 * (LW + WH + HL)
New Surface Area = 2 * (9LW + 9WH + 9HL)
By comparing the two surface area formulas, we can determine whether the new box will require three times as much cardboard.
If the expression for the new surface area is indeed three times larger than the original surface area, i.e., New Surface Area = 3 * Original Surface Area, then the new box will require three times as much cardboard. Otherwise, it won't.
Substituting the simplified equations:
2 * (9LW + 9WH + 9HL) = 3 * (2 * (LW + WH + HL))
Simplifying further:
18LW + 18WH + 18HL = 6LW + 6WH + 6HL
Combining like terms:
12LW + 12WH + 12HL = 0
Now, at this point, we can see that the equation is not satisfied, indicating that the new box will not require three times as much cardboard as the original box.