Problem 62 - Algebra A Combined Function

Question

Solve the following. For the solutions have been started for you. One side of a triangle is three times as long as another side, and the third side is 12 inches long. If the perimeter can be no longer than 32 inches, find the maximum lengths of the other two sides.

Step-by-Step Solution

Answer

The sides are 5 inches and 15 inches, with the third side 12 inches.

Step 1: Define Variables

Let's denote the sides of the triangle as follows:- Let the first side be denoted as \( x \).- The second side will then be \( 3x \) (since it is three times the first side).- The third side is given as 12 inches.

Step 2: Write the Perimeter Inequality

The perimeter of the triangle is given as the sum of its three sides. Therefore, the inequality for the perimeter can be written as: \[ x + 3x + 12 \leq 32 \] which simplifies to: \[ 4x + 12 \leq 32 \]

Step 3: Solve the Inequality

Subtract 12 from both sides of the inequality: \[ 4x + 12 - 12 \leq 32 - 12 \]This simplifies to: \[ 4x \leq 20 \]Next, divide each side by 4 to solve for \( x \): \[ x \leq 5 \].

Step 4: Calculate the Maximum Side Lengths

From the inequality \( x \leq 5 \), the maximum length for \( x \) is 5 inches. The second side, being three times the first side, has a maximum length of:\[ 3x = 3 \times 5 = 15 \] inches.Thus, when \( x \) is at its maximum, the side lengths are \( 5 \), \( 15 \), and \( 12 \) inches respectively.

Step 5: Verify the Solution

Check the solution to ensure it respects all conditions. Calculate the perimeter with \( x = 5 \):\[ 5 + 15 + 12 = 32 \] This satisfies the condition that the perimeter is no more than 32 inches. Hence, the solution is correct.

Key Concepts

Perimeter of a Triangle

The perimeter of a triangle is the total length around the triangle. To find it, we simply add up the lengths of all three sides.
In this problem, we are dealing with a triangle whose sides are defined as follows: one side is represented by the length \( x \), another by \( 3x \) (meaning it's three times longer than the first side), and the final side is 12 inches.
Thus, the perimeter equation is formed by adding these three lengths together:

• First side: \( x \)
• Second side: \( 3x \)
• Third side: 12 inches

These are summed up as:
\[ x + 3x + 12 \]
By simply solving this, you can determine if the perimeter constraint, in this case, less or equal to 32 inches, is met.

Inequalities in Algebra

Inequalities are expressions that show the relationship between quantities that are not equivalent. They are fundamentally important in algebra, as they help us understand how different values relate to one another.
In this exercise, we used an inequality to deal with the perimeter of the triangle. The inequality is written as:

• \( 4x + 12 \leq 32 \)

This tells us that the sum of \( 4x + 12 \) must be less than or equal to 32, meaning the perimeter cannot exceed 32 inches.
By subtracting 12 from each side, we aim to isolate the variable \( x \) and simplify the inequality:

• \( 4x + 12 - 12 \leq 32 - 12 \)
• Which simplifies to \( 4x \leq 20 \)

This action of manipulating the inequality is crucial in finding the limits imposed by the problem.

Solving Algebraic Equations

Solving algebraic equations involves finding the value of the variable that makes the equation true. In this context, our goal was to determine the maximum possible lengths of the triangle's sides.
We began by solving the simplified equation \( 4x \leq 20 \). To do this, divide each side of the equation by 4 to isolate \( x \):

• \( 4x \div 4 \leq 20 \div 4 \)
• This results in \( x \leq 5 \)

The maximum possible value for \( x \) is therefore 5. With the first side being 5 inches, the second side can now be calculated since it's three times \( x \):

• \( 3x = 3 \times 5 = 15 \) inches.

Finally, always verify your solutions by substituting back into the original conditions.
This ensures the perimeter is within the limit set by the problem, confirming that the side lengths \( 5 \), \( 15 \), and 12 inches indeed satisfy the inequality.