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Chapter 7: Problem 4

Find the constant of proportionality, \(k,\) for the given conditions. a. \(y=k x^{3},\) and \(y=64\) when \(x=2\). b. \(y=k x^{3 / 2},\) and \(y=96\) when \(x=16\). c. \(A=k r^{2},\) and \(A=4 \pi\) when \(r=2\). d. \(v=k t^{2},\) and \(v=-256\) when \(t=4\).

Short Answer

Expert verified
a) 8, b) 1.5, c) \(\pi\), d) -16

Step by step solution

01

Step 1a: Write the equation

Given the function is of the form \(y = k x^{3}\). Plug in the values for \(y\) and \(x\).
02

Step 2a: Substitute values

Substitute \(y = 64\) and \(x = 2\) into the equation: \(64 = k (2)^{3}\).
03

Step 3a: Solve for k

Solve the equation for \(k\). \[\begin{aligned} 64 & = k (8) \ k & = \frac{64}{8} \ k & = 8. \end{aligned}\]
04

Step 1b: Write the equation

Given the function is of the form \(y = k x^{3 / 2}\). Plug in the values for \(y\) and \(x\).
05

Step 2b: Substitute values

Substitute \(y = 96\) and \(x = 16\) into the equation: \(96 = k (16)^{3/2}\).
06

Step 3b: Solve for k

Solve the equation for \(k\). \[\begin{aligned} 96 & = k (64) \ k & = \frac{96}{64} \ k & = \frac{3}{2}. \end{aligned}\]
07

Step 1c: Write the equation

Given the function is of the form \(A = k r^{2}\). Plug in the values for \(A\) and \(r\).
08

Step 2c: Substitute values

Substitute \(A = 4 \pi\) and \(r = 2\) into the equation: \(4 \pi = k (2)^{2}\).
09

Step 3c: Solve for k

Solve the equation for \(k\). \[\begin{aligned} 4 \pi & = k (4) \ k & = \frac{4 \pi}{4} \ k & = \pi. \end{aligned}\]
10

Step 1d: Write the equation

Given the function is of the form \(v = k t^{2}\). Plug in the values for \(v\) and \(t\).
11

Step 2d: Substitute values

Substitute \(v = -256\) and \(t = 4\) into the equation: \(-256 = k (4)^{2}\).
12

Step 3d: Solve for k

Solve the equation for \(k\). \[\begin{aligned} -256 & = k (16) \ k & = \frac{-256}{16} \ k & = -16. \end{aligned}\]

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Key Concepts

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

Understanding Algebraic Equations
Algebraic equations are mathematical statements that show the equality between two expressions. They often contain variables, which represent unknown numbers. In our exercise, we explore forms like:
  • y = kx^3
  • y = kx^(3/2)
  • A = kr^2
  • v = kt^2
Here, we need to identify the variable 'k,' known as the constant of proportionality. Algebra helps us express relationships between quantities and solve for unknowns by manipulating equations using basic arithmetic operations.
Solving for k
To find the constant of proportionality, 'k,' you need to follow these steps: First, substitute the given values into the provided equation. Then isolate 'k' through algebraic manipulation. Here are our examples:
  • For the equation y = kx^3, if y = 64 and x = 2, substitute these values to get 64 = k(2)^3. Solve for 'k' to find k = 8.
  • For y = kx^(3/2), if y = 96 and x = 16, substitute these values to get 96 = k(16)^(3/2). Solve for 'k' to get k = 3/2.
  • Similarly, in A = kr^2, if A = 4蟺 and r = 2, substitute to get 4蟺 = k(2)^2. Solving for 'k' gives us k = 蟺.
  • Finally, for v = kt^2, with v = -256 and t = 4, we substitute to get -256 = k(4)^2. Solving for 'k' results in k = -16.
This process highlights substitution and solving for the unknown, key techniques in algebra.
Proportional Relationships
A proportional relationship implies two quantities increase or decrease at the same rate. Mathematically, if y is proportional to x, then y = kx, where 'k' is the constant of proportionality. In our problems, we analyzed different powers of 'x' and 'r.'
  • In y = kx^3, 'k' indicates how y changes with x cubed.
  • In y = kx^(3/2), 'k' reflects how y varies with x raised to the three-halves power.
  • For A = kr^2, 'k' shows the proportional change in area with respect to the square of the radius.
  • In v = kt^2, 'k' indicates the change of velocity as a function of the squared time.
Recognizing these relationships helps solve complex problems by understanding patterns and rates of change.
Exploring Exponential Functions
Exponential functions involve expressions where variables appear as exponents. In our context, we deal with functions like y = kx^n, where 'n' can be any real number. These functions grow rapidly, making them useful in various applications:
  • Finance, for calculating compound interest.
  • Biology, to model population growth.
  • Physics, for radioactive decay and more.
In our problems, we tackled forms like x^3 and x^(3/2), as well as quadratic forms like r^2 and t^2. The principle remains the same: understand the specific exponent to solve for the constant 'k' and comprehend the growth or decay in the context of the function.

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Most popular questions from this chapter

(Graphing program option.) For each part sketch by hand the three graphs on the same grid and clearly label each function. Describe how the three graphs are alike and not alike. a. \(y=x^{1} \quad y=x^{3} \quad y=x^{5}\) b. \(y=x^{2} \quad y=x^{4} \quad y=x^{6}\) c. \(y=x^{3} \quad y=2 x^{3} \quad y=-2 x^{3}\) d. \(y=x^{2} \quad y=4 x^{2} \quad y=-4 x^{2}\) If possible, check your results using a graphing program.

(Graphing program recommended.) If \(x\) is positive, for what values of \(x\) is \(3 \cdot 2^{x}<3 \cdot x^{2}\) ? For what values of \(x\) is \(3 \cdot 2^{x}>3 \cdot x^{2} ?\)

Solve the following formulas for the indicated variable. a. \(V=l w h,\) solve for \(l\) b. \(A=\frac{h b}{2},\) solve for \(b\) c. \(P=2 l+2 w\), solve for \(w\) d. \(S=2 x^{2}+4 x h,\) solve for \(h\)

Given the following three power functions in the form \(y=k x^{p}\) $$ y_{1}=x^{3} \quad y_{2}=5 x^{3} \quad y_{3}=2 x^{4} $$ a. Use the rules of logarithms to change each power function to the form: \(\log y=\log k+p \log x\). b. Substitute in each equation in part (a), \(Y=\log y\) and \(X=\log x\) and the value of \(\log k\) to obtain a linear function in \(X\) and \(Y\). c. Compare your functions in parts (a) and (b) to the original functions. What does the value of \(\log k\) represent in the linear equation? What does the value of the slope represent in the linear equation?

Given the functions \(f(x)=x^{4}\) and \(g(x)=(4)^{x}\) a. Find \(f(3 x)\) and \(3 f(x)\). Summarize the difference between these functions and \(f(x)\). b. Find \(g(3 x)\) and \(3 g(x)\). Summarize the difference between these functions and \(g(x)\).
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