Problem 20 - Algebra 1: Concepts and Skills

Question

A bottle-nosed whale can dive at a rate of 440 feet per minute. You want to find how long it will take for a bottle-nosed whale to dive 2475 feet at this rate. Which equation represents this situation? A. \(t=2475-440\) B. \(t=\frac{2475}{440}\)

Step-by-Step Solution

Answer

The correct equation to represent this situation is \(t=\frac{2475}{440}\)

Step 1: Understand the problem

The problem presents a situation where a whale dives at a rate of 440 feet per minute and you need to calculate how long it will take the whale to dive 2475 feet at this rate. According to the formula for distance (D=RT), to solve for time (T), divide the total distance (D) by the rate (R).

Step 2: Set up solution

Following Step 1, the equation that represents the situation is \(t=\frac{D}{R}\), where R is 440 feet per minute and D is 2475 feet. Therefore, the equation to solve this problem is \(t=\frac{2475}{440}\)

Step 3: Identify the correct answer

Therefore, the correct equation that represents this situation is \(t=\frac{2475}{440}\)

Key Concepts

Distance Formula

The distance formula is essential to solve many real-world problems involving movement and rates. It is often expressed as \( D = R \times T \), where \(D\) stands for distance, \(R\) for rate, and \(T\) for time. In our whale diving problem, the goal is to calculate the time it takes for the whale to dive a certain distance. This distance is 2475 feet, and it is covered at a constant rate of 440 feet per minute.
To find the time \(T\), we rearrange the formula to solve for \(T\). This gives us \(T = \frac{D}{R}\).
Here, we substitute 2475 feet for \(D\) and 440 feet per minute for \(R\), leading to the equation \(t = \frac{2475}{440}\).

• This type of problem illustrates how changes in distance and rate affect the time required for a journey or task.
• Understanding the relationship between these variables is crucial in figuring out travel times or rates of processes.

Solving Equations

Solving equations is a foundational skill in algebra that allows us to find unknown values, such as the time we wish to determine in the whale diving problem. Once we have set up our equation \(t = \frac{2475}{440}\), we proceed by performing the division.

The equation is simplified by dividing 2475 by 440 to find the value of \(t\).
By performing this calculation, one can easily determine the time it will take for the whale to complete the dive.

• The ability to manipulate equations like this one is critical for solving various mathematical problems in daily life.
• Understanding how to isolate a variable, like time \(t\) in our equation, is fundamental to problem-solving in mathematics.

By mastering the steps involved in solving this equation, students can confidently approach similar mathematical challenges.

Mathematical Modeling

Mathematical modeling is the process of representing real-world scenarios with mathematical forms to predict outcomes or understand relationships. In the case of the diving whale problem, mathematical modeling helps us visualize how distance, rate, and time interconnect.

Modeling begins with identifying the vital variables involved in the problem: in this case, the distance (2475 feet), the rate (440 feet per minute), and the time (\(t\)). By framing these elements with the formula \(t = \frac{D}{R}\), we are able to construct a clear path toward solving for time.

• Mathematical modeling translates complex real-world activities into manageable mathematical problems.
• It helps us make informed predictions and decisions based on quantifiable data.

Thus, integrating mathematical modeling in solving this problem showcases its value in problem-solving and highlights its applicability in diverse fields beyond mathematics itself.