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Chapter 7: Problem 21

Write each formula using the "language" of variation. For example, the formula for the circumference of a circle, \(C=2 \pi r,\) can be written as "The circumference of a circle varies directly as the length of its radius." \(S=4 \pi r^{2},\) where \(S\) is the surface area of a sphere with radius \(r\)

Short Answer

Expert verified
The surface area of a sphere varies directly as the square of the radius.

Step by step solution

01

Identify the variables

Identify which variables are involved in the formula. In this case, the formula is given by ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline
02

Determine the type of variation

Establish how the surface area ewline ewline ewline ewline ewline ewline ewline
03

Write the statement in the language of variation

Combine the information from steps ewline ewline ewline ewline

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Key Concepts

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

Direct Variation
Direct variation refers to a relationship between two variables where if one variable increases, the other variable increases at a constant rate. In other words, if variable y varies directly as variable x, then the formula can be written asy = kx, where k is the constant of proportionality.
For instance, if you were to increase the length of a radius in a circle, its circumference would also increase directly with it.
This principle is crucial in understanding numerous mathematical relationships and formulas.
It allows for a clear, proportional relationship to be established between various quantities.
Surface Area of a Sphere
The surface area of a sphere is the total area that covers the outer layer of the sphere. The formula used to calculate this is:
\(S = 4 \pi r^{2}\)
Here, S represents the surface area and r represents the radius of the sphere.
This equation tells us that the surface area is directly proportional to the square of its radius.
If you think of peeling an orange and laying out the peel flat, the size of that peel depends on how big the orange was.
Radius and Surface Area Relationship
The relationship between the radius and the surface area of a sphere is a classic example of direct variation.
As the radius increases, the surface area increases as well.
However, it's essential to note that this isn't a simple, linear relationship because the surface area increases with the square of the radius.
This means if you double the radius, the surface area increases by a factor of four (since 2 squared is 4).
This quadratic relation is fundamental in understanding how changes in one dimension affect multiple dimensional measurements.

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Most popular questions from this chapter

Determine whether each equation represents direct, inverse, joint, or combined variation. $$ y=\frac{4 x}{w z} $$

Forensic scientists use the lengths of certain bones to calculate the height of a person. Two bones often used are the tibia \((t),\) the bone from the ankle to the knee, and the femur \((r),\) the bone from the knee to the hip socket. A person's height ( \(h\) ) in centimeters is determined from the lengths of these bones by using functions defined by the following formulas. For men: \(h(r)=69.09+2.24 r\) or \(h(t)=81.69+2.39 t\) For women: \(h(r)=61.41+2.32 r\) or \(h(t)=72.57+2.53 t\) (a) Find the height of a man with a femur measuring \(56 \mathrm{cm}\). (b) Find the height of a man with a tibia measuring \(40 \mathrm{cm} .\) (c) Find the height of a woman with a femur measuring \(50 \mathrm{cm} .\) (d) Find the height of a woman with a tibia measuring \(36 \mathrm{cm}\).

For each function, find (a) \(f(2)\) and \((b) f(-1) .\) See Examples 4 and \(5 .\) $$ f=\\{(2,5),(3,9),(-1,11),(5,3)\\} $$

Solve each problem. The maximum load of a horizontal beam that is supported at both ends varies directly as the width and the square of the height and inversely as the length between the supports. A beam \(6 \mathrm{m}\) long, \(0.1 \mathrm{m}\) wide, and \(0.06 \mathrm{m}\) high supports a load of \(360 \mathrm{kg} .\) What is the maximum load supported by a beam \(16 \mathrm{m}\) long, \(0.2 \mathrm{m}\) wide, and \(0.08 \mathrm{m}\) high?

Find an equation of the line that satisfies the given conditions. See Example 4. Through \((2,-7) ;\) horizontal
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