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Chapter 7: Problem 1

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 \(h\)

Short Answer

Expert verified
a. \(l = \frac{V}{w h}\), b. \(b = \frac{2A}{h}\), c. \(w = \frac{P - 2l}{2} \), d. \(h = \frac{S - 2 x^{2}}{4 x}\)

Step by step solution

01

- Solve for l in the formula for volume

Given the formula for volume, \(V=l w h\), isolate \(l\) by dividing both sides of the equation by \(w h\).\[l = \frac{V}{w h}\]
02

- Solve for b in the formula for area

Given the formula for the area of a triangle, \(A=\frac{h b}{2}\), solve for \(b\) by multiplying both sides of the equation by 2 and then dividing by \(h\).\[b = \frac{2A}{h}\]
03

- Solve for w in the formula for perimeter

Given the formula for the perimeter, \(P=2 l+2 w\), isolate \(w\) by first subtracting \(2 l\) from both sides, then dividing by 2.\[w = \frac{P - 2l}{2} \]
04

- Solve for h in the formula for surface area

Given the formula for surface area, \(S=2 x^{2}+4 x h\), isolate \(h\) by subtracting \(2 x^{2}\) from both sides, then dividing by \(4 x\).\[h = \frac{S - 2 x^{2}}{4 x}\]

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Key Concepts

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

volume formula
Understanding the volume formula is crucial in solving many geometry and physics problems. The volume of a rectangular prism can be calculated using the formula: \( V = lwh \). Here,
  • \(V\) represents the volume,
  • \(l\) represents the length,
  • \(w\) represents the width,
  • \(h\) represents the height.
To solve for the length (\(l\)), you rearrange the equation by dividing both sides by \(wh\): \[ l = \frac{V}{wh} \]. This way, you can find out the length if you know the volume, width, and height.
area formula of a triangle
The area of a triangle is given by the formula \(A = \frac{bh}{2}\). Here,
  • \(A\) is the area,
  • \(b\) is the base,
  • \(h\) is the height.
If you need to solve for the base (\(b\)), you can rearrange the formula. Multiply both sides by 2 to get \(2A = bh\), and then divide by \(h\) to isolate \(b\): \[ b = \frac{2A}{h} \]. This formula helps you find the base if you have the area and the height.
perimeter formula
The perimeter of a rectangle can be found using \(P = 2l + 2w\). In this formula,
  • \(P\) is the perimeter,
  • \(l\) is the length,
  • \(w\) is the width.
If you need to solve for the width (\(w\)), start by subtracting \(2l\) from both sides to get \(P - 2l = 2w\). Then, divide by 2: \[ w = \frac{P - 2l}{2} \]. Now, you can find the width if you know the perimeter and the length.
surface area formula
The surface area of a solid can often require more complex formulas. For a rectangular solid, a useful formula is \(S = 2x^2 + 4xh\).
  • \(S\) is the surface area,
  • \(x\) is often a given dimension,
  • \(h\) is the height.
To solve for the height (\(h\)), first isolate \(h\) by subtracting \(2x^2\) from both sides: \(S - 2x^2 = 4xh\). Then divide by \(4x\): \[ h = \frac{S - 2x^2}{4x} \]. This equation is handy for determining height when the surface area and another dimension are known.

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Most popular questions from this chapter

Write a general formula to describe each variation. Use the information given to find the constant of proportionality. a. \(Q\) is directly proportional to both the cube root of \(t\) and the square of \(d\), and \(Q=18\) when \(t=8\) and \(d=3\). b. \(A\) is directly proportional to both \(h\) and the square of the radius, \(r\), and \(A=100 \pi\) when \(r=5\) and \(h=2\). c. \(V\) is directly proportional to \(B\) and \(h,\) and \(V=192\) when \(B=48\) and \(h=4\) d. \(T\) is directly proportional to both the square root of \(p\) and the square of \(u,\) and \(T=18\) when \(p=4\) and \(u=6\).

Consider a cylinder with volume \(V=\pi r^{2} h\). What happens to its volume when you double its height, \(h\) ? When you double its radius, \(r\) ?

a. \(B\) is inversely proportional to \(x^{4}\). What is the effect on \(B\) of doubling \(x\) ? b. \(Z\) is inversely proportional to \(x^{p}\), where \(p\) is a positive integer. What is the effect on \(Z\) of doubling \(x\) ?

If \(f(x)=\frac{1}{x^{4}}\) and \(g(x)=\frac{1}{x^{5}},\) construct the following functions. a. \(f(-x)\) and \(-f(x) \quad\) c. \(g(-x)\) and \(-g(x)\) b. \(2 f(x)\) and \(f(2 x)\) d. \(2 g(x)\) and \(g(2 x)\) e. The function whose graph is the reflection of \(f(x)\) across the \(x\) -axis f. The function whose graph is the reflection of \(g(x)\) across the \(x\) -axis

When installing Christmas lights on the outside of your house, you read the warning "Do not string more than four sets of lights together." This is because the electrical resistance, \(R,\) of wire varies directly with the length of the wire, \(l,\) and inversely with the square of the diameter of the wire, \(d\). a. Construct an equation for electrical wire resistance. b. If you double the wire diameter, what happens to the resistance? c. If you increase the length by \(25 \%\) (say, going from four to five strings of lights), what happens to the resistance?
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