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Chapter 1: Problem 27

Research project: For this project you are to find and describe a function that is commonly used. Find a patient person whose job is interesting to you. Ask that person what types of calculations he or she makes. These calculations could range from how many bricks to order for building a wall to lifetime wages lost for a wrongful-injury settlement to how much insulin to inject. Be creative and persistentdon't settle for "I look it up in a table." Write a description, in words, of the function and how it is calculated. Then write a formula for the function, carefully identifying variables and units.

Short Answer

Expert verified
An architect uses the formula \( C = \frac{1}{6} \times f \times b \times d^2 \) to calculate beam load capacity.

Step by step solution

01

Selecting an Interesting Job

Choose a profession that fascinates you. For instance, you might select an architect because the calculations they perform have a direct impact on the structure and design of buildings.
02

Identifying the Calculation

Talk to the person in your chosen profession. For example, an architect may need to calculate the load-bearing capacity of a building's beam. This calculation directly influences the safety and integrity of the structure.
03

Describing the Calculation in Words

Describe the process of the calculation in simple words. An architect may determine the load capacity of a beam by considering the total weight that will be placed upon it and the material strength of the beam itself.
04

Defining the Function Formula

Write a formula for the function. For example, the load-bearing capacity (C) of a rectangular beam might be calculated as: \[ C = rac{1}{6} imes f imes b imes d^2 \]where \( f \) is the material strength in N/m虏, \( b \) is the width of the beam in meters, and \( d \) is the depth of the beam in meters.
05

Identifying the Variables and Units

Identify each variable and its unit clearly, such as: - \( f \): Material strength (N/m虏), - \( b \): Width of the beam (meters), - \( d \): Depth of the beam (meters).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Description
In the world of architectural engineering, a crucial function is determining the load-bearing capacity of structural elements like beams. Simply put, this is about understanding how much weight a beam can support without failing. Consider an architect who uses this function to ensure that every beam in their design can handle the weight it needs to support. They calculate the maximum weight a beam can hold by taking its material strength and size into account. This function is fundamental because it directly impacts both the safety and functionality of a building structure.
Variable Identification
When diving into the mechanics of a calculation, identifying variables is crucial. Variables are the symbols used to represent the different quantities that influence the calculation. In architectural calculations, such as finding the load-bearing capacity of a beam, three primary variables are usually identified:
  • Material Strength (f): Often measured in Newtons per square meter (N/m虏).
  • Width of the Beam (b): Measured in meters, it affects the cross-sectional area that contributes to the strength.
  • Depth of the Beam (d): Also in meters, greater depth often increases the capacity due to a larger moment of inertia.
By clearly identifying these variables and their units, one can ensure accurate and meaningful calculations.
Formula Development
Developing a formula involves creating an equation that connects all relevant variables to find a solution to a specific problem. For architectural calculations, the formula for the load-bearing capacity of a beam is expressed as:\[ C = \frac{1}{6} \times f \times b \times d^2 \]- Here, the function calculates capacity (C) using known values for material strength, width, and depth. Each part of the formula relies on empirical data from engineering studies about material properties and geometric principles. This formula helps translate architectural visions into real-world structures by ensuring beams can support their intended loads.
Architectural Calculations
Architectural calculations encompass a broad range of mathematical functions essential for designing safe and efficient structures. They translate architectural concepts into concrete numbers ensuring feasibility and safety. These calculations are vital during the design phase, influencing structural elements like beams, columns, and footings. For instance, ensuring a beam can bear stress affects not only safety but also the material costs and design aesthetics. By mastering these calculations, architects can deliver structures that are not only visually appealing but robust and reliable, safeguarding buildings from excessive stress and unforeseen failures.

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Most popular questions from this chapter

Arterial blood flow: Medical evidence shows that a small change in the radius of an artery can indicate a large change in blood flow. For example, if one artery has a radius only \(5 \%\) larger than another, the blood flow rate is \(1.22\) times as large. Further information is given in the table below. $$ \begin{array}{|c|c|} \hline \text { Increase in radius } & \begin{array}{c} \text { Times greater blood } \\ \text { flow rate } \end{array} \\ \hline 5 \% & 1.22 \\ \hline 10 \% & 1.46 \\ \hline 15 \% & 1.75 \\ \hline 20 \% & 2.07 \\ \hline \end{array} $$ a. Use the average rate of change to estimate how many times greater the blood flow rate is in an artery that has a radius \(12 \%\) larger than another. b. Explain why if the radius is increased by \(12 \%\) and then we increase the radius of the new artery by \(12 \%\) again, the total increase in the radius is \(25.44 \%\). c. Use parts a and \(b\) to answer the following question: How many times greater is the blood flow rate in an artery that is \(25.44 \%\) larger in radius than another? d. Answer the question in part c using the average rate of change.

Swimming records: The world record time for a certain swimming event was \(63.2\) seconds in 1950 . Each year thereafter, the world record time decreased by \(0.4\) second. a. Use a formula to express the world record time as a function of the time since 1950 . Be sure to explain the meaning of the letters you choose and the units. b. Express using functional notation the world record time in the year 1955 , and then calculate that value. c. Would you expect the formula to be valid indefinitely? Be sure to explain your answer.

Reynolds number: The Reynolds number is very important in such fields as fluid flow and aerodynamics. In the case of a fluid flowing through a pipe, the Reynolds number \(R\) is given by $$ R=\frac{v d D}{\mu} $$ Here \(v\) is the velocity of the fluid in meters per second, \(d\) is the diameter of the pipe in meters, \(D\) is the density of the fluid in kilograms per cubic meter, and \(\mu\) is the viscosity of the fluid measured in newton- seconds per square meter. Generally, when the Reynolds number is above 2000 , the flow becomes turbulent, and rapid mixing occurs. \({ }^{10}\) When the Reynolds number is less than 2000 , the flow is streamline. Consider a fluid flowing through a pipe of diameter \(0.05\) meter at a velocity of \(0.2\) meter per second. a. If the fluid in the pipe is toluene, its viscosity is \(0.00059\) newton- seconds per square meter, and its density is 867 kilograms per cubic meter. Is the flow turbulent or streamline? b. If the toluene is replaced by glycerol, then the viscosity is \(1.49\) newton-seconds per square meter, and the density is \(1216.3\) kilograms per cubic meter. Is the glycerol flow turbulent or streamline?

Darcy's law: The French hydrologist Henri Darcy discovered that the velocity \(V\) of underground water is proportional to the magnitude of the slope \(S\) of the water table (see Figure 1.71). The constant of proportionality in this case is the permeability of the medium through which the water is flowing. This proportionality relationship is known as Darcy's law, and it is important in modern hydrology. a. Using \(K\) as the constant of proportionality, express Darcy's law as an equation. b. Sandstone has a permeability of about \(0.041 \mathrm{me}-\) ter per day. If an underground aquifer is seeping through sandstone, and if the water table drops \(0.03\) vertical meter for each horizontal meter, what is the velocity of the water flow? Be sure to use appropriate units for velocity and keep all digits. c. Sand has a permeability of about 41 meters per day. If the aquifer from part a were flowing through sand, what would be its velocity?

How much can I borrow? The function in Example \(1.2\) can be rearranged to show the amount of money \(P=P(M, r, t)\), in dollars, that you can afford to borrow at a monthly interest rate of \(r\) (as a decimal) if you are able to make \(t\) monthly payments of \(M\) dollars: $$ P=M \times \frac{1}{r} \times\left(1-\frac{1}{(1+r)^{r}}\right) . $$ Suppose you can afford to pay \(\$ 350\) per month for 4 years. a. How much money can you afford to borrow for the purchase of a car if the prevailing monthly interest rate is \(0.75 \%\) ? (That is \(9 \%\) APR.) Express the answer in functional notation, and then calculate it. b. Suppose your car dealer can arrange a special monthly interest rate of \(0.25 \%\) (or \(3 \%\) APR). How much can you afford to borrow now? c. Even at \(3 \%\) APR you find yourself looking at a car you can't afford, and you consider extending the period during which you are willing to make payments to 5 years. How much can you afford to borrow under these conditions?
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