a school is planning to install new flooring in a classroom with an area of 80 square meters. they have two…

a school is planning to install new flooring in a classroom with an area of 80 square meters. they have two types of flooring to choose from: vinyl flooring, which costs $15 per square meter, and hardwood flooring, which costs $25 per square meter. the school must cover at least 30 square meters with hardwood flooring for durability purposes. how should the school allocate the flooring types to minimize costs, and what would the total cost be? use the keypad to enter your answer in the box. to minimize the cost, the school should cover square meters with vinyl flooring and square meters with hardwood flooring. the minimum cost of the entire project would be dollars.

a school is planning to install new flooring in a classroom with an area of 80 square meters. they have two types of flooring to choose from: vinyl flooring, which costs $15 per square meter, and hardwood flooring, which costs $25 per square meter. the school must cover at least 30 square meters with hardwood flooring for durability purposes. how should the school allocate the flooring types to minimize costs, and what would the total cost be? use the keypad to enter your answer in the box. to minimize the cost, the school should cover square meters with vinyl flooring and square meters with hardwood flooring. the minimum cost of the entire project would be dollars.

Answer

Explanation:

Step1: Define variables

Let $x$ be the area of vinyl flooring and $y$ be the area of hardwood flooring. We know that $x + y=80$ (total area) and $y\geq30$. The cost function $C = 15x+25y$.

Step2: Express $x$ in terms of $y$

From $x + y=80$, we get $x = 80 - y$.

Step3: Substitute $x$ into the cost - function

$C(y)=15(80 - y)+25y=1200-15y + 25y=1200 + 10y$.

Step4: Determine the value of $y$ for minimum cost

Since $y\geq30$ and $C(y)=1200 + 10y$ is a linear function with a positive slope ($m = 10>0$), the cost $C(y)$ is minimized when $y$ takes the smallest value that satisfies the constraint. So, $y = 30$.

Step5: Calculate $x$

If $y = 30$, then $x=80 - 30=50$.

Step6: Calculate the minimum cost

$C=15\times50+25\times30=750 + 750=1500$.

Answer:

To minimize the cost, the school should cover 50 square meters with vinyl flooring and 30 square meters with hardwood flooring. The minimum cost of the entire project would be 1500 dollars.