Question

Carolyn is 38 inches tall and is growing \(2 \frac{1}{2}\) inches per year. She wants to ride a roller coaster, but park rules set the minimum height at 48 inches. In. years, Carolyn will be tall enough to ride the roller coaster. $$ \begin{array}{|l|l|} \hline \text { Select.... } & \nabla \\ \hline 3 \\ \hline 4 \\ \hline 5 \\ \hline 6 \\ \hline \end{array} $$

Short answer

Carolyn will be tall enough to ride the roller coaster in 4 years.

Step-by-step solution

Understand the Problem

Carolyn's current height is 38 inches, and she needs to reach a height of 48 inches to ride a roller coaster. Determine how many years, growing at a rate of 2.5 inches per year, will allow her to achieve this height.

Set Up the Equation

Let the number of years required be represented by the variable, y. Carolyn's future height can be represented by the equation: \[ \text{Future Height} = \text{Current Height} + (\text{Growth Rate} \times \text{Years}) \] Substitute the given values into the equation: \[ 48 = 38 + 2.5y \]

Solve for y

Rearrange the equation to solve for y: \[ 48 - 38 = 2.5y \] \[ 10 = 2.5y \] Divide both sides by 2.5: \[ y = \frac{10}{2.5} \] \[ y = 4 \]

Verify the Solution

Substitute y = 4 back into the original equation to confirm: \[ 48 = 38 + (2.5 \times 4) \] \[ 48 = 38 + 10 \] \[ 48 = 48 \] The solution is verified.

Key concepts

Growth Calculation

Growth calculation is important for understanding how a quantity changes over time. In Carolyn's case, we need to figure out how much she will grow each year. She grows 2.5 inches annually. To determine her height after a certain number of years, multiply the growth rate by the number of years. For example, if Carolyn grows for 4 years, that's:
\(2.5 \times 4 = 10\) inches.
Growth calculations are common in many real-life problems, allowing us to predict future outcomes based on consistent rates of change. They're essential for making educated decisions about the future.

Linear Equations

Linear equations help us solve for unknown variables. In Carolyn's problem, we used the equation for her future height:
\(48 = 38 + 2.5y\)
First, isolate the variable:
\(48 - 38 = 2.5y\)
Linear equations are straightforward—they relate variables in a linear manner without exponents or complex functions. This makes them relatively simple to work with. Steps to solve a linear equation include:

• Isolate the variable term
• Simplify the equation
• Solve for the variable

Once you have the equation set up correctly, solving it involves basic arithmetic—adding, subtracting, multiplying, or dividing.

Height Requirements

Height requirements are common in many settings, such as amusement parks. These requirements ensure safety and appropriateness of rides. For Carolyn, the minimum height to ride the roller coaster is 48 inches. To figure out when she'll reach this height, we compared her growth rate with the requirement.
First, determine her future height using her growth rate. Then, check if it meets or exceeds the height requirement. In this problem, Carolyn's height after 4 years would be:
\(38 + 10 = 48\) inches.
Meeting height requirements ensures safety, as rides are designed to accommodate certain body sizes to prevent injuries. Thus, understanding and calculating when certain criteria are met is crucial.