Problem 20
Question
What will be the cost of installing a telephone line \(274 \mathrm{km}\) long, at \(\$ 5723\) per kilometer?
Step-by-Step Solution
Verified Answer
\$ 1,568,102
1Step 1: Understand the problem
You need to calculate the total cost of installing a telephone line. To do this, you multiply the cost per kilometer by the total number of kilometers.
2Step 2: Set up the equation
The cost per kilometer is given as \(\$ 5723\). To find the total cost, use the equation \(\text{Total cost} = \text{Cost per kilometer} \times \text{Total kilometers}\).
3Step 3: Calculate the total cost
Multiply the given cost per kilometer, \(\$ 5723\), by the total distance, \(274 \mathrm{km}\), to get the total cost.\[\text{Total cost} = \$ 5723 \times 274 \mathrm{km}\]
4Step 4: Perform the multiplication
Carry out the multiplication to find the total cost.\[\text{Total cost} = \$ 5723 \times 274 = \$ 1,568,102\]
Key Concepts
Mathematical Problem SolvingBasic AlgebraWord Problems in Mathematics
Mathematical Problem Solving
Mathematical problem solving is an essential skill that enables us to work through challenges by applying various mathematical concepts and techniques. When faced with mathematical word problems, like calculating costs, the first step is always to understand the question before jumping into calculations. This involves identifying what information is provided and what is being asked.
In the exercise about calculating the cost of installing a telephone line, the problem is straightforward and requires a simple multiplication. However, problem solving often involves more complex situations where several steps might be necessary to arrive at the solution. Effective mathematical problem solving includes creating a plan (which might involve setting up an equation or choosing a formula), carrying out the plan (performing operations), and then looking back to check if the solution makes sense.
To guide students in this process, it's useful to encourage them to always express the problem in their own words and to identify all relevant information before proceeding to solve it. Encouraging the breaking down of the problem into smaller, more manageable parts can also be quite beneficial.
In the exercise about calculating the cost of installing a telephone line, the problem is straightforward and requires a simple multiplication. However, problem solving often involves more complex situations where several steps might be necessary to arrive at the solution. Effective mathematical problem solving includes creating a plan (which might involve setting up an equation or choosing a formula), carrying out the plan (performing operations), and then looking back to check if the solution makes sense.
To guide students in this process, it's useful to encourage them to always express the problem in their own words and to identify all relevant information before proceeding to solve it. Encouraging the breaking down of the problem into smaller, more manageable parts can also be quite beneficial.
Basic Algebra
At the heart of many mathematical problems, including cost calculations, sits basic algebra. Algebra involves the use of symbols and letters to represent numbers and quantities in formulae and equations. The beauty of algebra is that it provides a systematic way of solving problems and is an indispensable tool in a variety of fields.
In the context of our exercise, algebra is utilized to set up the equation for calculating the total cost. The equation \(\text{Total cost} = \text{Cost per kilometer} \times \text{Total kilometers}\) is an algebraic expression that simplifies the process of computation. To enhance students' understanding, explain the meaning and importance of each term in the equation, and relate it to real-world context.
For instance, highlight that in algebra, we use variables to represent unknowns, which in this case are known values (cost per kilometer and total kilometers) used to find a specific end (total cost). Additionally, illustrating how changing one variable affects the overall solution can deepen students' appreciation of algebra's utility in problem solving.
In the context of our exercise, algebra is utilized to set up the equation for calculating the total cost. The equation \(\text{Total cost} = \text{Cost per kilometer} \times \text{Total kilometers}\) is an algebraic expression that simplifies the process of computation. To enhance students' understanding, explain the meaning and importance of each term in the equation, and relate it to real-world context.
For instance, highlight that in algebra, we use variables to represent unknowns, which in this case are known values (cost per kilometer and total kilometers) used to find a specific end (total cost). Additionally, illustrating how changing one variable affects the overall solution can deepen students' appreciation of algebra's utility in problem solving.
Word Problems in Mathematics
Word problems in mathematics bridge the gap between abstract mathematical concepts and real-world situations. They require students to translate text-based information into mathematical expressions—a critical skill in day-to-day problem solving. The exercise we're discussing is an example of a word problem where the student is required to extract numerical information and perform calculations based on it.
Encoding a word problem into a mathematical equation is a two-fold task. First, identify the numerical information given: in our case, the cost per kilometer and the total number of kilometers. Second, associate these numbers with the correct mathematical operations to find the desired outcome, which is the total cost.
To improve student's ability to tackle word problems, encourage them to visualize the scenario, for example by drawing a diagram or a picture. Emphasize the importance of understanding common keywords that indicate mathematical operations, like 'per' suggesting multiplication in this context. Lastly, it can be useful to practice rewriting word problems in simpler language to fully grasp the question before attempting a solution.
Encoding a word problem into a mathematical equation is a two-fold task. First, identify the numerical information given: in our case, the cost per kilometer and the total number of kilometers. Second, associate these numbers with the correct mathematical operations to find the desired outcome, which is the total cost.
To improve student's ability to tackle word problems, encourage them to visualize the scenario, for example by drawing a diagram or a picture. Emphasize the importance of understanding common keywords that indicate mathematical operations, like 'per' suggesting multiplication in this context. Lastly, it can be useful to practice rewriting word problems in simpler language to fully grasp the question before attempting a solution.
Other exercises in this chapter
Problem 20
Convert between the given customary and metric units. 2.55 horsepower to kilowatts.
View solution Problem 20
Reciprocals Find the reciprocal of each number, retaining the proper number of digits in your answer. $$-6.93$$
View solution Problem 20
Determine the number of significant digits in each approximate number. $$1.0000$$
View solution Problem 20
Evaluate each expression. Retain the proper number of significant digits in your answer. Powers by Calculator. $$(1.94)^{7}$$
View solution