Question

The cost of a square slab is proportional to its thickness and also proportional to the square of its length. What is the cost of a square slab that is 3 meters long and 0.1 meter thick?

(1) The cost of a square slab that is 2 meters long and 0.2 meter thick is $160 more than the cost of a square slab that is 2 meters long and 0.1 meter thick.

(2) The cost of a square slab that is 3 meters long and 0.1 meter thick is $200 more than the cost of a square slab that is 2 meters long and 0.1 meter thick.

Answer 1

(1) The cost of the square slab is $360

(2) The cost of the square slab is $360

Step-by-step explanation:

Cost (C) varies as thickness (t) and square of length (L^2)

Therefore, C = ktL^2

Case 1

C1 = 160 + C2

C1 - C2 = 160

C1 = ktL^2 (t = 0.2, L = 2)

C1 = k×0.2×2^2 = 0.8k

C2 = ktL^2 (t = 0.1, L = 2)

C2 = k×0.1×2^2 = 0.4k

C1 - C2 = 0.8k - 0.4k = 0.4k

C1 - C2 = 0.4k

160 = 0.4k

k = 160/0.4 = 400

C = ktL^2 (t = 0.1, t = 3)

C = 400×0.1×3^2 = $360

Case 2

C1 = 200 + C2

C1 - C2 = 200

C1 = ktL^2 (t = 0.1, L = 3)

C1 = k×0.1×3^2 = 0.9k

C2 = ktL^2 (t = 0.1, L = 2)

C2 = k×0.1×2^2 = 0.4k

C1 - C2 = 0.9k - 0.4k = 0.5k

C1 - C2 = 0.5k

200 = 0.5k

k = 200/0.5 = 400

C = ktL^2 (t = 0.1, L = 3)

C = 400×0.1×3^2 = $360

Answer 2

$520

Step-by-step explanation:

Using information provided in the question, equations can be formed to determine the unknown proportionality factors.

`C=AT + BL^2` , where C is the cost, T is the thickness, L is the length and A and B are the proportionality factors

Equation 1:

Slab 1:
`C_1=0.2A + 2^2B=0.2A+4B`

Slab 2:
`C_2=0.1A + 2^2B=0.1A+4B`

`C_1=C_2+160`

`0.2A+4B=0.1A+4B+160`

`0.1A=160`

`A=1600`

Equation 2:

Slab 2:
`C_2=0.1A + 3^2B=0.1A+9B`

Slab 3:
`C_3=0.1A + 2^2B=0.1A+4B`

`C_2=C_3+200`

`0.1A+9B=0.1A+4B+200`

`5B=200`

`B=40`

After determining unknowns A=1600 and B=40, these can be substituted into
`C=AT + BL^2` to give below

`C=1600T + 40L^2`

Cost is to be determined for T=0.1m and L=3m

`C=1600T + 40L^2`

`C=1600(0.1) + 40(3)^2`

`C=160 + 360`

`C=520`

The cost of a square slab of 3 meter length and 0.1 meter thickness is $520