Question

A bottle-nosed whale can dive 440 feet per minute. Suppose a bottle-nosed whale is 500 feet deep and dives at this rate. Write and solve an equation to find how long it will take to reach a depth of 2975 feet. Round to the nearest whole minute.

Short answer

It will take approximately 6 minutes for the whale to reach a depth of 2975 feet.

Step-by-step solution

Understand the Problem

The whale is initially at a depth of 500 feet and can dive an additional 440 feet per minute. The problem is to find out how long, in minutes, will it take for the whale to reach a depth of 2975 feet.

Formulate the Equation

An equation that will determine the time taken (t) by the whale to reach a depth of 2975 feet can be established as follows: 500 + 440t = 2975

Solve the Equation

Subtract 500 from both sides of the equation to isolate terms with the variable on one side: 440t = 2975 - 500, which simplifies to 440t = 2475. Now, divide both sides of the equation by 440 to solve for t. So, t = 2475/440.

Round the Answer

Since the problem asks for the answer to the nearest whole minute, round the value obtained in the previous step to the nearest whole number. It gives t = 6 minutes.

Key concepts

Linear Equations

A linear equation is like a balance scale where both sides need to remain equal. In the context of the whale diving problem, we are discussing a linear equation in the form of initial depth + (rate of diving × time) = final depth.

Linear equations are usually written in the form ax + b = c, where a, b, and c are numbers, and x represents the variable we want to solve for. The aim is to find the value of x which makes the equation true. To achieve this, we primarily perform operations such as adding, subtracting, multiplying, or dividing both sides of the equation by the same number, to isolate x and find its value.

When solving a linear equation in a real-world context, like the whale problem, the same principles apply. We isolated the variable t, representing time, by subtracting the initial depth from the final depth and then dividing by the diving rate to get the answer.

Word Problems in Algebra

Approaching word problems in algebra can often seem intimidating, but taking the right steps can simplify the process. First, we need to read the problem carefully and determine what we're being asked to find—in this case, it's the time it will take for the whale to reach a certain depth. Next, identify and assign variables to the unknown quantities and use the given information to formulate an equation.

In the whale diving problem, we pinpointed that the depth increases by a certain rate per minute, which provided us the clue to set up the equation. Breaking down the problem into smaller parts and translating them into mathematical language is key. The focus should always be on understanding the scenario and the relationship between different quantities given in the problem before attempting to solve it.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. In our whale example, 440t is an algebraic expression. Here, t represents the variable time, and 440 is the coefficient that corresponds to the rate of diving in feet per minute.

The beauty of algebraic expressions is that they can model real-world situations, such as the distance a whale dives over time. Understanding how to manipulate these expressions according to the rules of algebra is essential to solving equations. For instance, we solved the whale problem by isolating the variable t after performing operations like subtraction and division on the original algebraic expression. Recognizing and efficiently manipulating these expressions is a fundamental skill in algebra that applies to numerous real-life problems.