Question

when preparing for meals at a banquet hall, the chef calculates he will need $2x^5$ chicken dishes, $3x^2$ fish dishes and $5x$ vegetarian dishes per banquet room. he knows 4 of the banquet rooms will be used that night.

1. write and simplify the expression to show how many total dishes the chef needs that night.

1. if $x = 3$, how many total of each type of dish will there be?

chicken ____ fish __ vegetarian ____

Explanation:

Sub - question 1

Step 1: Find dishes per room sum

First, find the total number of dishes per banquet room by adding the number of chicken, fish, and vegetarian dishes. So, per room, it's \(2x^{5}+3x^{2}+5x\).

Step 2: Multiply by number of rooms

Since there are 4 banquet rooms, we multiply the per - room total by 4. Using the distributive property \(a(b + c + d)=ab+ac + ad\), we get \(4(2x^{5}+3x^{2}+5x)=4\times2x^{5}+4\times3x^{2}+4\times5x\).

Step 3: Simplify the expression

Simplify each term: \(4\times2x^{5} = 8x^{5}\), \(4\times3x^{2}=12x^{2}\), and \(4\times5x = 20x\). So the simplified expression is \(8x^{5}+12x^{2}+20x\).

Step 1: Calculate chicken dishes

For chicken dishes, the formula per night is \(4\times2x^{5}=8x^{5}\). Substitute \(x = 3\) into the formula: \(8\times3^{5}\). First, calculate \(3^{5}=3\times3\times3\times3\times3 = 243\). Then, \(8\times243=1944\).

Step 2: Calculate fish dishes

For fish dishes, the formula per night is \(4\times3x^{2}=12x^{2}\). Substitute \(x = 3\) into the formula: \(12\times3^{2}\). Calculate \(3^{2}=9\), then \(12\times9 = 108\).

Step 3: Calculate vegetarian dishes

For vegetarian dishes, the formula per night is \(4\times5x=20x\). Substitute \(x = 3\) into the formula: \(20\times3=60\).

Answer:

The expression for the total number of dishes is \(8x^{5}+12x^{2}+20x\).

Sub - question 2