carl has a box with a length width and height of 12 inches. she needs a box that is 4 inches longer, not as high, yet still has exactly the same width and volume.
what are the new dimensions?
which box requires more material to construct?
PLEASE HELP ME!! :(
length would be 16 , width 12, and hight 9 .... but you should double check
how did you figure that out and its correct it has the same volume and width
new box would be 12 by 16 by x
12(16)(x) = original volume = 12^3 = 1728
x = 1728/(12(16) ) = 9
so they both share an equal material?
no, they have the same volume, but different surface areas (material used)
in original, you would have 6 equal squares
each of 12 by 12 or 144 inches^2
so the surface area of the box including top and bottom = 6(144) or 864 square inches
surface area of the new box
= top and bottom + fron and back + two ends
= 2(16 x 12) + 2(16 x 9) + 2( 12 x 9) = 888 square inches
Make a sketch to see how the calculations developed.
To find the new dimensions of the box, we need to follow the given instructions: the new box should be 4 inches longer, not as high, but still have the same width and volume.
Let's start by finding the volume of Carl's original box. The volume of a rectangular box is calculated by multiplying its length, width, and height. In this case, all three dimensions are 12 inches, so the volume is:
Volume of original box = length * width * height
= 12 inches * 12 inches * 12 inches
= 1,728 cubic inches
Now, we need to find the new dimensions of the box by increasing the length by 4 inches and keeping the width the same. To keep the volume the same, the new height needs to be adjusted accordingly.
Let the new length be L.
Since the length is increased by 4 inches, we have L = 12 inches + 4 inches = 16 inches.
Let the new height be H.
Since the new box should have the same volume, we can use the formula for volume and solve for H.
Volume of new box = length * width * height
1,728 cubic inches = 16 inches * 12 inches * H
H = 1,728 cubic inches / (16 inches * 12 inches)
H ≈ 9 inches (rounded to the nearest whole number)
Therefore, the new dimensions of the box are:
Length = 16 inches
Width = 12 inches (same as the original box)
Height = 9 inches
To determine which box requires more material to construct, we need to calculate the surface area of each box. The surface area of a rectangular box is calculated by finding the areas of each face and adding them together.
Let's calculate the surface area of Carl's original box:
Surface area of original box = 2 * (length * width + width * height + height * length)
= 2 * (12 inches * 12 inches + 12 inches * 12 inches + 12 inches * 12 inches)
= 2 * (144 square inches + 144 square inches + 144 square inches)
= 2 * 432 square inches
= 864 square inches
Now, let's calculate the surface area of the new box:
Surface area of new box = 2 * (length * width + width * height + height * length)
= 2 * (16 inches * 12 inches + 12 inches * 9 inches + 9 inches * 16 inches)
= 2 * (192 square inches + 108 square inches + 144 square inches)
= 2 * 444 square inches
= 888 square inches
Therefore, the new box requires more material to construct because it has a larger surface area of 888 square inches compared to the original box's surface area of 864 square inches.