Surface Area of Composite Solids

Geometric Shape	Surface Area B = area of the base P = perimeter of the base	Volume B = area of the base P = perimeter of the base
prism (general)	SA = 2B + Ph	V = Bh
triangular prism	SA = 2B + Ph SA = 2(1/2ab) + (b + c + d)h SA = ab + (b + c + d)h	V = Bh V = 1/2 abh
rectangular prism	SA = 2B + Ph SA = 2(lw) + (2l + 2w)h	V = lwh
Regular square prism	SA = 2B + Ph SA = 2(s2) + (4s)h	V = Bh V = s2h
regular pentagonal prism	SA = 2B + Ph SA = 2(1/2ans) + nsh SA = 2(1/2a)(5)s + 5sh SA = 5as + 5sh	V = Bh V = 1/2ansh V = 1/2a(5)sh V = 5/2ash
regular hexagonal prism	SA = 2B + Ph SA = 2(1/2ans) + nsh SA = 2(1/2a)(6)s + 6sh SA =6as + 6sh	V = Bh V = 1/2ansh V = 1/2a(6)sh V = 3ash
cube	SA = 2B + Ph SA = 2(s2) + (4s)s = 6s2	V = Bh V = s3
regular pyramid (general)	SA = B + n(1/2sl) l = slant height	V = 1/3Bh
regular triangular pyramid	SA = B + n(1/2sl) SA = 1/2as + (3)(1/2sl) SA = 1/2as + 3/2sl l = slant height	V = 1/3Bh V = 1/3(1/2 ab)h V = 1/6 abh
regular square pyramid	SA = B + n(1/2sl) SA = s2 + (4)(1/2sl) SA = s2 + 2sl l = slant height	V = 1/3Bh V = 1/3(b2)h = 1/3b2h
regular pentagonal pyramid	SA = B + n(1/2sl) SA = 1/2a(5)s + (5)(1/2sl) SA = 5/2as + 5/2sl l = slant height	V = 1/3Bh V = 1/3(1/2anb)h V = 1/6 anbh
regular hexagonal pyramid	SA = B + n(1/2sl) SA = 1/2a(6)s + (6)(1/2sl) SA = 3as + 3sl l = slant height	V = 1/3Bh V = 1/3(1/2anb)h V = 1/6 anbh
cylinder	SA = 2B + ph SA = 2(p r2) + (2pr)h	V = Bh V = p r2h

Some suggestions of your project work's answers

PART I:

Get the information from internet, related it to the mathematics field such as Geometry, Progressions, Calculus, and so on.

PART II:

1) To find the diameter of baking tray to be used to fit the 5 kg cake (1 kg = 3800 cm^3), all you need is the formula of volume of cylinder, V = pi x (d/2)^2 x h. You should get the answer 58.78 cm.

2) (a) To find the corresponding values of diameters of the baking tray to be used, all you need is the formula that related h and d, which is the formula of volume of cylinder, V = pi x (d/2)^2 x h, and you are given that the weight of the cake must be 5kg, so you should able to find it's volume since 1 kg = 3800 cm^3.

(b) (i) To find the range of heights that is NOT suitable for the cakes, you just have to think logically, the height and the diameter of the cake should not exceed the inner dimensions of the oven.

(ii) The answer is not fixed, you just have to explain why you choose that dimension.

(c) This question is related to Linear Law, since the question asks you to form an equation to represent the LINEAR relation between h and d. You must form the linear equation by reducing the equation V = pi x (d/2)^2 x h, where the V already given, in cm^3. Your Y and X must consist of h and/or d, and your m and c only consist of constant, but not h and/or d. After that plot your line of best fit.

(ii) (a)  Just refer to your line of best fit to find the diameter of the round cake pan required where height of the cake is 10.5. Just do as what normally we do, just don't forget that the value you get from the line of best fir is either Y/X!! You have to calculate again to get your d!


(ii) (b) Same as (ii) (a), the only different is you need to find h by referring to the d = 42 cm

3) (a) By referring to your 2(b)(ii), you can estimate the amount of fresh cream just by using the formula of total surface, A = pi x r^2 + 2 x pi x r x h. The original formula of total surface area of a cylinder should be A = 2 x pi x r^2 + 2 x pi x r x h, but please think logically, we are not going to put any cream at the BOTTOM of the cake, so we we only need to consider the top surface area of the cake.You should able to find the amount (volume, in cm^3) of fresh cream required to decorate the cake, where the thickness of the cream is 1 cm.

(b) Suggest three other shapes for cake besides cylinder. You may refer to the shapes such as cube, cuboid, triangular prism, pentagon-shape based solid, and so on. Different shape will have different way to find the total surface area. You should able to find the amount (volume, in cm^3) of fresh cream required to decorate the cake, where the thickness of the cream is 1 cm.

(c) By referring to the volume of cream required for the cylinder cake and the other three different shapes of cake suggested in (b), determine the shape requires the least amount of fresh cream to be used.

PART III:

The method that you may refer are differentiation (Calculus) and Completing the Square from Quadratic Functions. The question asks for the dimension of a 5 kg round cake that requires the minimum amount of fresh cream, from here you should know (i) minimum total surface area will requires minimum amount of fresh cream, (ii) the weight of the cake already fixed at 5 kg, this means the volume of cake also fixed (1kg = 3800 cm^3), (iii) The height, h and the diameter, d of the cake will give you the dimension of the cake, so you need to derive an equation of total surface area (not involved the bottom of the cake) in terms of h or d/r (r = d/2), so that later on you can find the values of them.

Volume, V =pi x r^2 x h

Total surface area (not include bottom of cake), A = pi x r^2 + 2 x pi x r x h

Differentiation:

You may get the equation of A in terms of h or d/r here (better A in terms of d/r, and you should able to get A = pi/4 x d^2 + 76000/d  or  A = pi x r^2 + 38000/r if you refer to radius, r).
If referring to A = pi x r^2 + 38000/r, dA/dr = 0 when total surface area is minimum, from here you should able to get the values of r, and then the corresponding values of d (d = 2r).

Completing the Square:

If referring to A = pi x r^2 + 2 x pi x r x h, factorize the pi so that a = 1, you will get A = pi (r^2 + 2hr), then do the completing the square to find the r in terms of h, after that substitute it inside V =pi x r^2 x h to get the value of h and r (negative value will be rejected, impossible).

After this, determine whether you want to bake the cake with this dimension or not, explain your decision.

FURTHER EXPLORATION:

(a) This question is related to progression, because the radius of the next cake will be 10% less than the previous cake, determine whether this is AP or GP, and then determine the a and d/r so that you can find the volume of the first, second, third and fourth cakes.

(b) Total mass of all cakes (Sn) not exceed (<) 15 kg, Sn < 15, and you need the formula of sum of terms. After you get the answer, verify your answer by other methods, you may use the listing method, list down one by one.

REFLECTION:

Based on your own creativity.

CONCLUSION:

Briefly conclude everything that you have done in this project.

Ganbate!!!