Question

Kelp Growth If kelp grows at a steady rate of \(30 \mathrm{~cm}\) per day, how long would it take to reach a length of \(25 \mathrm{~m}\) ?

Short answer

It takes approximately 83.33 days for the kelp to reach 25 meters.

Step-by-step solution

Understand the Problem

The problem asks how many days it will take for the kelp to grow to a length of 25 meters if it grows at a rate of 30 centimeters per day. We need to determine the number of days required to achieve this length.

Convert Units

First, convert the length from meters to centimeters, because the growth rate is given in centimeters per day. Since 1 meter equals 100 centimeters, \[ 25 ext{ meters} = 25 imes 100 ext{ cm} = 2500 ext{ cm} \]

Set Up the Equation

To find the number of days, use the equation \( ext{days} = \frac{ ext{total length in cm}}{ ext{growth rate per day in cm}} \). Substitute the known values:\[ ext{days} = \frac{2500 ext{ cm}}{30 ext{ cm/day}} \]

Calculate the Number of Days

Perform the division to find the number of days:\[ ext{days} = \frac{2500}{30} = 83.ar{3} \] This means it would take approximately 83.33 days.

Key concepts

Unit Conversion

Converting units is a crucial step in solving many algebraic problems, as it ensures all quantities are in the same measurement system. In this particular exercise, we needed to convert the length of kelp from meters to centimeters because the growth rate was provided in centimeters per day. Here's how you can approach unit conversion:

• Identify the units you are converting from and to. In this case, meters to centimeters.
• Use a conversion factor, which is a number that converts one unit to another. Here, 1 meter = 100 centimeters.
• Apply the conversion by multiplying the original measure by the conversion factor. For 25 meters, this means doing 25 × 100, giving us 2500 centimeters.

By ensuring consistent units, the calculations are much easier and more accurate. It's a simple yet powerful tool in problem solving.

Rate of Growth

Understanding the rate of growth is important in this exercise because it allows us to predict how long it will take for something to reach a certain size or length. The kelp’s growth rate in this problem is 30 centimeters per day. This tells us that each day, the kelp will uniformly increase by 30 centimeters.

• A rate of growth describes how quickly something changes over time.
• It can be thought of as a ratio or fraction, like 30 cm/day.
• The rate of growth remains constant in this problem, meaning it doesn't change over time, making your calculations straightforward.

By understanding the growth rate, we can efficiently determine how long a process, like kelp growing, will take.

Mathematical Equations

A mathematical equation is a statement that expresses the equality of two expressions. In this exercise, we set up an equation to find out how many days it would take for the kelp to reach a specified length. We use the formula: \[ \text{days} = \frac{\text{total length in cm}}{\text{growth rate per day in cm}} \]The equation is a practical tool that bridges the known values (total length and growth rate) to the unknown (number of days). Here are the steps to set up this equation:

• Identify what you know and what you need to find out. Here we know the total length in centimeters and the rate at which the kelp grows.
• Write the equation that relates the known variables with the unknown variable using division, as it’s the operation that solves how many groups (or days) fit into a total (length).
• Substitute the known values into the equation. This problem shows 2500 cm for the total length and 30 cm/day as the growth rate.

Division and Approximation

Division is the mathematical operation used to find out how many times one quantity is contained within another. After setting up our equation, we performed division to determine the number of days. Using:\[ \text{days} = \frac{2500}{30} = 83.\overline{3} \]We find that the exact number of days it will take for the kelp to grow to 25 meters is approximately 83.33 days. When dealing with decimals in division:

• Consider rounding for simplicity, especially when the exact decimal is impractical. Rounding can simplify your result to a more digestible number.
• In this case, 83.33 was approximated to 83.33 days for a more precise answer instead of rounding to a whole number.
• Approximations are useful when perfect precision isn’t necessary for practical purposes.

Thus, division and approximation help in arriving at an understandable and useful solution.