Maddy’s loaf tin has three partitioned areas in which to bake three loaves. The outside of the tin along with the partitions is made of 80 inches of material. The area of the entire loaf tin is 192 in2. What are the possible outer dimensions of the loaf tin?
if the length is x and the width and partitions is y, then
2x+4y = 80
x+2y = 40
xy = 192
So, which pairs factors of 192 add up as needed?
1 192
2 96
3 64
4 48
and so on. In the list you will find the pair(s) you need.
Let's assume the outer dimensions of the loaf tin are length (L), width (W), and height (H).
The outside of the tin, excluding the partitions, forms a rectangular box with dimensions (L - 2H), (W - 2H), and H.
The area of this rectangular box is given by the formula: 2(L - 2H)(W - 2H) + 2(L - 2H)H + 2(W - 2H)H = 80 in².
The area of the entire loaf tin, including the partitions, is given by the formula: (L - 2H)(W - 2H) = 192 in².
To find the possible values of L, W, and H, let's solve these two equations simultaneously.
From equation 1: 2(L - 2H)(W - 2H) + 2(L - 2H)H + 2(W - 2H)H = 80 in²
Expanding and simplifying: 2(LW - 4LH - 4WH + 8H²) + 2LH - 4H² + 2WH - 4H² = 80 in²
Simplifying further: 2LW - 8LH - 8WH + 16H² + 2LH - 4H² + 2WH - 4H² = 80 in²
Combining like terms: 2LW - 6LH - 10H² + 2WH = 80 in²
From equation 2: (L - 2H)(W - 2H) = 192 in²
Expanding: LW - 2LH - 2WH + 4H² = 192 in²
Now we have a system of two equations:
2LW - 6LH - 10H² + 2WH = 80 -----(eq. 3)
LW - 2LH - 2WH + 4H² = 192 -----(eq. 4)
To solve this system, we can either use substitution or elimination method.
Let's use the substitution method:
Rearrange eq. 4 to express LW in terms of H: LW = 2LH + 2WH - 4H² + 192
Substitute this value of LW in eq. 3:
2(2LH + 2WH - 4H² + 192) - 6LH - 10H² + 2WH = 80
Simplify: 4LH + 4WH - 8H² + 384 - 6LH - 10H² + 2WH = 80
Combine like terms: -2LH - 16H² + 6WH = -304
Rearrange: 3WH - 2LH - 16H² = -304 -----(eq. 5)
Now we have eq. 5 and eq. 3 as:
3WH - 2LH - 16H² = -304 -----(eq. 5)
2LW - 6LH - 10H² + 2WH = 80 -----(eq. 3)
Now, we have a system of two equations. Solving these equations will give us the possible values for L, W, and H.
To find the possible outer dimensions of the loaf tin, we need to first consider the total area of the outside and partitions.
Let's assume the dimensions of the loaf tin as follows:
- Length (L)
- Width (W)
- Height (H)
Since the loaf tin has three partitioned areas, we can assume that the length and width of each partition is L/3 and W/3 respectively.
The outside of the tin can be calculated by adding the area of the base and the combined area of all four sides. The base area is L * W, while the area of each side can be found by multiplying the height (H) with the sum of the length and width of the partitioned area (L/3 + W/3).
So, the total area of the outside and partitions is calculated as:
Total area = base area + 4 * (side area)
Total area = L * W + 4 * (H * (L/3 + W/3))
Given that the total area of the outside and partitions is 80 inches, we can set up the equation as follows:
80 = L * W + 4 * (H * (L/3 + W/3))
Also, given that the area of the entire loaf tin is 192 in², we can set up another equation as follows:
192 = L * W
Now, we can solve these two equations simultaneously to find the possible values for the outer dimensions of the loaf tin.