Problem 54 - College Algebra

Question

A prospective buyer wants to know how much grain a specific silo can hold. The area of the floor of the silo is \((2 x+9)^{2}\) . The height of the silo is \(10 x+10\) , where \(x\) is measured in feet. Expand the square and multiply by the height to find the expression that shows how much grain the silo can hold.

Step-by-Step Solution

Answer

The volume is \(40x^3 + 400x^2 + 1170x + 810\).

Step 1: Expand the Square of the Floor Area

The expression for the floor area of the silo is given by \((2x + 9)^2\). We need to expand this square. Recall that \((a + b)^2 = a^2 + 2ab + b^2\). Here, \(a = 2x\) and \(b = 9\), so: \(\ (2x + 9)^2 = (2x)^2 + 2(2x)(9) + 9^2 = 4x^2 + 36x + 81\).

Step 2: Find Volume of the Silo

The volume of a silo is given by the formula: \(\text{Volume} = \text{Area of the Base} \times \text{Height}\). We have expanded the area of the base to be \(4x^2 + 36x + 81\). The height of the silo is given as \(10x + 10\). So, the volume \(V\) becomes \((4x^2 + 36x + 81)(10x + 10)\).

Step 3: Multiply the Expanded Area by the Height

Now, expand the expression \((4x^2 + 36x + 81)(10x + 10)\):1. Multiply \(4x^2\) by \(10x + 10\): \(4x^2 \cdot 10x = 40x^3\) \(4x^2 \cdot 10 = 40x^2\)2. Multiply \(36x\) by \(10x + 10\): \(36x \cdot 10x = 360x^2\) \(36x \cdot 10 = 360x\)3. Multiply \(81\) by \(10x + 10\): \(81 \cdot 10x = 810x\) \(81 \cdot 10 = 810\)4. Sum all the parts together: \(V = 40x^3 + 40x^2 + 360x^2 + 360x + 810x + 810\) 5. Combine like terms: \(V = 40x^3 + (40x^2 + 360x^2) + (360x + 810x) + 810\) \(= 40x^3 + 400x^2 + 1170x + 810\).

Step 4: Final Solution

The final expanded expression for the volume of the silo, which tells us how much grain it can hold, is: \(40x^3 + 400x^2 + 1170x + 810\).

Key Concepts

Polynomial Expansion

Polynomial expansion is a fundamental algebraic technique used to simplify expressions, particularly when dealing with expressions raised to a power such as - \((a+b)^2 = a^2 + 2ab + b^2\). This expansion helps to express the floor area of the silo in simplified polynomial form.

When expanding \((2x + 9)^2\), we use:

• Identify the terms: \(a = 2x\) and \(b = 9\).
• Use the identity: \((a + b)^2 = a^2 + 2ab + b^2\)
• Plug them into the identity to get: \((2x)^2 + 2(2x)(9) + 9^2\)
• Results in \(4x^2 + 36x + 81\)

This expression is crucial for calculating the base area, making it easy to further compute the volume of the silo.

Volume Calculation

Finding the volume of a shape like a cylindrical silo involves multiplying the area of its base by its height. The formula used is:- \[ \text{Volume} = \text{Area of the Base} \times \text{Height}\] Here, the base area was calculated as \(4x^2 + 36x + 81\) using polynomial expansion.

The height of the silo is given as \(10x + 10\).To find the volume, substitute these values:

• Volume, \(V = (4x^2 + 36x + 81)(10x + 10)\)
• This results in a polynomial expression after applying the distributive property.

Expanding this expression gives a clear demonstration of how algebraic skills are employed practically to solve real-life problems.

Algebraic Expressions

Algebraic expressions are used extensively to represent real-world problems in mathematical terms. An algebraic expression consists of numbers, variables, and operators. For instance, \((2x + 9)^2\) represents the area of the silo's base.

In this problem, we expanded the expression and then used multiplication to find the volume:

• Combine terms systematically: starting from squared terms \(4x^2\) and ending with constants \(81\)
• Once they are expanded, they can be multiplied across, multiplying each term by each term from the other polynomial expression.

This guides us to the final result:\(40x^3 + 400x^2 + 1170x + 810\),providing clarity and a step-by-step approach to handling lengthy calculations through algebra. Working with algebraic expressions helps in visualizing how components of a problem interact and aids in developing problem-solving strategies.