The problem describes Han building a dog house with a budget of $65. He needs at least 60 square feet of plywood, at least 36 feet of wood planks, and between 1 and 2 quarts of paint. The costs are $0.70 per square foot of plywood, $0.10 per foot of wood planks, and $8 per quart of paint. The task is to write inequalities to represent the material and cost constraints.

1. Problem Description

The problem describes Han building a dog house with a budget of $

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5. He needs at least 60 square feet of plywood, at least 36 feet of wood planks, and between 1 and 2 quarts of paint. The costs are $0.70 per square foot of plywood, $0.10 per foot of wood planks, and $8 per quart of paint. The task is to write inequalities to represent the material and cost constraints.

2. Solution Steps

First, we define the variables:

Let $p$ be the number of square feet of plywood.

Let $w$ be the number of feet of wood planks.

Let $q$ be the number of quarts of paint.

The material constraints are:

* Plywood: At least 60 square feet, so $p \ge 60$.

* Wood planks: At least 36 feet, so $w \ge 36$.

* Paint: Between 1 and 2 quarts, so $1 \le q \le 2$.

The cost constraint is:

The total cost must be no more than $

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5. The cost of plywood is $0.70p$.

The cost of wood planks is $0.10w$.

The cost of paint is $8q$.

So, the total cost is $0.70p + 0.10w + 8q$, which must be less than or equal to $

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5.

Thus, the cost constraint is $0.70p + 0.10w + 8q \le 65$.

3. Final Answer

The inequalities representing the material and cost constraints are:

$p \ge 60$

$w \ge 36$

$1 \le q \le 2$

$0.70p + 0.10w + 8q \le 65$

Where:

$p$ = square feet of plywood

$w$ = feet of wood planks

$q$ = quarts of paint