Question

Math

Posted 5 months ago

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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?
```

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Answer from Sia

Posted 5 months ago

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.

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