Answer:
The length of the base is 8 cm.
Explanation:
To find the length of the base of the box, we can use the formula for the total surface area of a box. The formula is:
Surface Area = 2lw + 2lh + 2wh
where l is the length, w is the width, and h is the height of the box.
In this case, we are given that the height of the box is 6cm and the total surface area is 128cm².
Using the formula, we can substitute the given values:
128 = 2lw + 2(6)(l) + 2w(6)
Next, let's simplify the equation:
128 = 2lw + 12l + 12w
Now, we can solve for the length of the base by rearranging the equation:
2lw + 12l + 12w - 128 = 0
To make it easier to solve, let's divide the equation by 2:
lw + 6l + 6w - 64 = 0
Now, we can factor the equation:
(l + 6)(w + 6) - 64 = 0
Next, we can rearrange the equation to isolate one variable:
(l + 6)(w + 6) = 64
Now, we need to find two numbers that multiply to 64 and have a sum of 6. By trial and error, we find that the numbers are 8 and 8.
So, we have:
(l + 8)(w + 8) = 64
Now, we can set each factor equal to zero and solve for l:
l + 8 = 0
l = -8
Since the length of a base cannot be negative, we can discard this solution.
Therefore, the length of the base of the box is 8 cm.
To summarize:
- The height of the box is 6 cm.
- The total surface area is 128 cm².
- By using the formula for the total surface area of a box and solving the resulting equation, we find that the length of the base is 8 cm.