a sheet of cardboard with dimensions 12cm by 10cm will have square corners of x cm cut from it. The cardboard will then be folded to form a box.
1. explain why the length and breadth of the box will be 12 - 2x and 10 - 2x respectively.
2. find the expanded expression for the external surface area of the cardboard box
3. find an expanded expression for the volume of the box
1. show, completing all steps, that the area of the entire garden bed and pathway is 6x^2 + 28x + 16m^2
2. write an expression for the area of the new pathway
+128475
1. We start off with a length of 12 and a breadth of 10...so.....if we cut "x" off of both ends of these, each will be decreased by 2x. so the new legth will be 12 - 2x and 10 - 2x.
2. The surface area of the bottom willl be L x W = (12 - 2x)(10 - 2x) = 120 + 4x^2 - 44x.
The surface area of the sides will be....
2x [(12 - 2x) + (10 - 2x)] = 2x [ 22 - 4x] = 4x [11 - 2x]. = 44x - 8x^2.
So. adding these together...the expression for the total surface area is ... -4x^2 +120
3. The volume will be L x W x H = (12 - 2x)* (10 - 2x) *x = [120 + 4x^2 - 44x] * x = 4x^3 - 44x^2 + 120x
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1. Since we're adding 2 ft on each side, the total area is ...
(6x + 2 + 2) (x+ 2 + 2) = (6x + 4) (x + 4) = [6x^2 + 28x + 16 ] m^2
2. The area of the pathway is just the total area - the area of the garden =
[6x^2 + 28x + 16] - [(6x * x] = [6x^2 + 28x + 16] - (6x^2] = 28x + 16
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+128475
1. We start off with a length of 12 and a breadth of 10...so.....if we cut "x" off of both ends of these, each will be decreased by 2x. so the new legth will be 12 - 2x and 10 - 2x.
2. The surface area of the bottom willl be L x W = (12 - 2x)(10 - 2x) = 120 + 4x^2 - 44x.
The surface area of the sides will be....
2x [(12 - 2x) + (10 - 2x)] = 2x [ 22 - 4x] = 4x [11 - 2x]. = 44x - 8x^2.
So. adding these together...the expression for the total surface area is ... -4x^2 +120
3. The volume will be L x W x H = (12 - 2x)* (10 - 2x) *x = [120 + 4x^2 - 44x] * x = 4x^3 - 44x^2 + 120x
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1. Since we're adding 2 ft on each side, the total area is ...
(6x + 2 + 2) (x+ 2 + 2) = (6x + 4) (x + 4) = [6x^2 + 28x + 16 ] m^2
2. The area of the pathway is just the total area - the area of the garden =
[6x^2 + 28x + 16] - [(6x * x] = [6x^2 + 28x + 16] - (6x^2] = 28x + 16
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