Fake gold bricks are made by covering concrete cubes with gold paint, so the cost of the paint is proportional
to their surface area while the cost of the concrete is proportional to their volume. If a 1 inch cube costs $1.30 to make while a 2 inch cube
costs $6.80, then how much would a 3 inch cube cost?

Question

Fake	gold	bricks	are	made	by	covering	concrete	cubes	with	gold	paint,	so	the	cost	of	the	paint	is	proportional	
to	their	surface	area	while	the	cost	of	the	concrete	is	proportional	to	their	volume.		If	a	1	inch	cube	costs	$1.30	to	make	while	a	2	inch	cube	
costs	$6.80,	then	how	much	would	a	3	inch	cube	cost?

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To find the cost of a 3-inch cube, we first need to understand the relationship between the cost, surface area, and volume for the given cubes. ‖ ‖ step 2 ⋮ Let's denote the cost to make a cube with side length $ s $ inches as $ C(s) $, the surface area of the cube as $ A(s) = 6s^2 $, and the volume of the cube as $ V(s) = s^3 $. ‖ ‖ step 3 ⋮ The cost for a 1-inch cube is $ C(1) = \$1.30 $, and for a 2-inch cube is $ C(2) = \$6.80 $. ‖ ‖ step 4 ⋮ We can express the cost as $ C(s) = k_1A(s) + k_2V(s) $, where $ k_1 $ is the cost per unit surface area for the paint, and $ k_2 $ is the cost per unit volume for the concrete. ‖ ‖ step 5 ⋮ Using the costs for 1-inch and 2-inch cubes, we can set up a system of equations to solve for $ k_1 $ and $ k_2 $. ‖ ‖ step 6 ⋮ Substituting $ A(1) = 6 $ and $ V(1) = 1 $ into $ C(1) = k_1A(1) + k_2V(1) $, we get $ 1.3 = 6k_1 + k_2 $. ‖ ‖ step 7 ⋮ Substituting $ A(2) = 24 $ and $ V(2) = 8 $ into $ C(2) = k_1A(2) + k_2V(2) $, we get $ 6.8 = 24k_1 + 8k_2 $. ‖ ‖ step 8 ⋮ Solving the system of equations from steps 6 and 7, we find the values of $ k_1 $ and $ k_2 $. ‖ ‖ step 9 ⋮ Once we have $ k_1 $ and $ k_2 $, we can calculate the cost for a 3-inch cube using $ C(3) = k_1A(3) + k_2V(3) $, where $ A(3) = 54 $ and $ V(3) = 27 $. ‖ ‖ step 10 ⋮ Substitute $ k_1 $ and $ k_2 $ into the equation for $ C(3) $ to find the final cost. ‖ ∻Answer∻ ⚹ [Insert final answer here] ⚹ ∻Key Concept∻ ⚹Cost Proportionality to Surface Area and Volume⚹ ∻Explanation∻ ⚹The cost of making the cubes is proportional to their surface area and volume, which means we can express the cost as a linear combination of the surface area and volume with constant coefficients. By solving the system of equations derived from the known costs of smaller cubes, we can find these coefficients and use them to calculate the cost of a larger cube.⚹◊What is the surface area formula of a cube?⍭ Generate me a similar question◊

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