Here is the question:
Consider the following rectangular piece of tinplate. An open top cake tin is to be made by cutting a square (xcm by xcm) from each corner.
*sorry, I wasn't sure how to insert the picture so I attempted to draw it.
Develop a conjecture about the relationship between x (the cut to be made for the square) and the length of each side of the rectangle such that the cake tin has a maximum volume.
The sides of the rectangle are in a ratio r:s. Consider a rectangle where one side is twice the length of the other (i.e. 2:1). Find the value of x that gives the maximum volume for the cake tin. Repeat this process for rectangular tin plates with sides in at least two other ratios.
Hint: find exact solutions for x (i.e. use the quadratic formula).
My progress so far:
I have been able to find the equation for the volume, which is:
V(x) = x(L-2x)(W-2x)
After substituting the values for L and W (1 and 2 respectively for the 1:2 ratio asked in the question), I found the derivative, which was:
12x^2 - 12x +2
From this, I used the quadratic formula to find the x values, which were:
x = 0.789 and x = 0.211
Since 0.789 is not in the domain and would result in a negative volume,
I concluded that 0.211 was the x value of the maximum volume.
I repeated this with ratios of 1:3, 1:4 and 1:5 and found these maximum x values:
1:3 : x = 0.2257
1:4 : x = 0.2324
1:5 : x = 0.2362
The question asked me to develop a conjuncture about the relationship between x and the length of each side of the rectangle. This section I was unsure about. I attempted to, but I wasn't able to find any relation between the x value and the length of r and s (if there is one). If someone could help with this it would be much appreciated, thanks.
Consider the following rectangular piece of tinplate. An open top cake tin is to be made by cutting a square (xcm by xcm) from each corner.
Code:
+-+--------||-------+-+
| |x x| |
+-+ +-+
|x x|
| |
+-+ +-+
| |x x| |
+-+--------||-------+-+
*sorry, I wasn't sure how to insert the picture so I attempted to draw it.
Develop a conjecture about the relationship between x (the cut to be made for the square) and the length of each side of the rectangle such that the cake tin has a maximum volume.
The sides of the rectangle are in a ratio r:s. Consider a rectangle where one side is twice the length of the other (i.e. 2:1). Find the value of x that gives the maximum volume for the cake tin. Repeat this process for rectangular tin plates with sides in at least two other ratios.
Hint: find exact solutions for x (i.e. use the quadratic formula).
Code:
dimensions:
|<-----r cm---------->|___
+-+--------||-------+-+ ^
| |x x| | |
+-+ +-+ |
|x x| s cm
| | |
+-+ +-+ |
| |x x| | |
+-+--------||-------+-+_v_
My progress so far:
I have been able to find the equation for the volume, which is:
V(x) = x(L-2x)(W-2x)
After substituting the values for L and W (1 and 2 respectively for the 1:2 ratio asked in the question), I found the derivative, which was:
12x^2 - 12x +2
From this, I used the quadratic formula to find the x values, which were:
x = 0.789 and x = 0.211
Since 0.789 is not in the domain and would result in a negative volume,
I concluded that 0.211 was the x value of the maximum volume.
I repeated this with ratios of 1:3, 1:4 and 1:5 and found these maximum x values:
1:3 : x = 0.2257
1:4 : x = 0.2324
1:5 : x = 0.2362
The question asked me to develop a conjuncture about the relationship between x and the length of each side of the rectangle. This section I was unsure about. I attempted to, but I wasn't able to find any relation between the x value and the length of r and s (if there is one). If someone could help with this it would be much appreciated, thanks.
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