Part B (Square Cake Tins): Let I (cm) be the side length of the square tinplate and let x (cm) be the side length of the square cuts to be made. (a) V(x) = 4x3 – 40x² + 100x cm³. Hence, determine V (x) and find the exact value of x which will maximise the volume of the cake tin. Given l = 10cm, show that the volume of the cake tin can be expressed as (b) Further this investigation by determining the exact value of x for at least two other values of I cm (side length of square tinplate). Present a conjecture based on a square piece of tinplate of side length l cm, which when a square is (c) cut from each corner of length x cm, a maximum volume for the resulting open-top cake tin will be obtained. Show sufficient working to support your conjecture. Part C (Rectangular Cake Tins): Consider the following rectangular piece of tinplate. An open-top cake tin is to be made by cutting a square from each corner. The sides of the rectangular tinplate are in a ratio p: q. (a) Consider a rectangle where one side is twice the length of the other (i.e. p: q = 2:1). Using your process and findings from Part B, determine the exact value of x that gives the maximum volume for this cake tin. (b) Repeat this process for a rectangular tinplate in the ratio 3:1. (c) Explore rectangular tinplate with sides in at least two other ratios. Hint: Change the value of both p and q, and determine exact solutions for x. (d) 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 tinplate (p and q) such that the cake tin has a maximum volume. Show sufficient working to support your conjecture.
Part B (Square Cake Tins): Let I (cm) be the side length of the square tinplate and let x (cm) be the side length of the square cuts to be made. (a) V(x) = 4x3 – 40x² + 100x cm³. Hence, determine V (x) and find the exact value of x which will maximise the volume of the cake tin. Given l = 10cm, show that the volume of the cake tin can be expressed as (b) Further this investigation by determining the exact value of x for at least two other values of I cm (side length of square tinplate). Present a conjecture based on a square piece of tinplate of side length l cm, which when a square is (c) cut from each corner of length x cm, a maximum volume for the resulting open-top cake tin will be obtained. Show sufficient working to support your conjecture. Part C (Rectangular Cake Tins): Consider the following rectangular piece of tinplate. An open-top cake tin is to be made by cutting a square from each corner. The sides of the rectangular tinplate are in a ratio p: q. (a) Consider a rectangle where one side is twice the length of the other (i.e. p: q = 2:1). Using your process and findings from Part B, determine the exact value of x that gives the maximum volume for this cake tin. (b) Repeat this process for a rectangular tinplate in the ratio 3:1. (c) Explore rectangular tinplate with sides in at least two other ratios. Hint: Change the value of both p and q, and determine exact solutions for x. (d) 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 tinplate (p and q) such that the cake tin has a maximum volume. Show sufficient working to support your conjecture.
Chapter5: Polynomial And Rational Functions
Section: Chapter Questions
Problem 6PT: Solve the following application problem. A rectangular field is to be enclosed by fencing. In...
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Transcribed Image Text:Part B (Square Cake Tins):
Let I (cm) be the side length of the square tinplate and let x (cm) be the side length of the square cuts to be
made.
(a)
V(x) = 4x3 – 40x² + 100x cm³.
Hence, determine V (x) and find the exact value of x which will maximise the volume of the cake tin.
Given l = 10cm, show that the volume of the cake tin can be expressed as
(b)
Further this investigation by determining the exact value of x for at least two other values of I cm
(side length of square tinplate).
Present a conjecture based on a square piece of tinplate of side length l cm, which when a square is
(c)
cut from each corner of length x cm, a maximum volume for the resulting open-top cake tin will be
obtained. Show sufficient working to support your conjecture.
Part C (Rectangular Cake Tins):
Consider the following rectangular piece of tinplate. An open-top cake tin is to be made by cutting a square
from each corner. The sides of the rectangular tinplate are in a ratio p: q.
(a)
Consider a rectangle where one side is twice the length of the other (i.e. p: q = 2:1). Using your
process and findings from Part B, determine the exact value of x that gives the maximum volume for
this cake tin.
(b)
Repeat this process for a rectangular tinplate in the ratio 3:1.
(c)
Explore rectangular tinplate with sides in at least two other ratios.
Hint: Change the value of both p and q, and determine exact solutions for x.
(d) 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 tinplate (p and g) such that the cake tin has a maximum volume.
Show sufficient working to support your conjecture.
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