A formula is a mathematical relationship that is expressed in symbols (variables).
You may recognize the following:
\(P=2l+2w\)
\(A=lw\)
\(A=\pi r^2\)
\(V=\pi r^2h\)
The formula \(P=2l+2w\) is used to determine the perimeter of a rectangle given length, \(l\), and width, \(w\).
Isolate \(w\):
\(P\)
\(=2w + 2l\)
\(P-2l\)
\(=2w\)
\(\dfrac{P-2l}{2}\)
\(=w\)
\(\dfrac{P}{2}-\dfrac{2l}{2}\)
\(\dfrac{P}{2}-l\)
Approach 1 — Use the original equation
\(2l+2w\)
\(=P\)
\(2(5)+2w\)
\(=36\)
\(10+2w\)
\(2w\)
\(=26\)
\(w\)
\(=\dfrac{26}{2}\)
\(=13\)
Approach 2 — Use the rearranged equation
\( \dfrac{36}{2}-5\)
\(18-5\)
\(13\)
For a length of \(8\)
\(\begin{align*} \frac{P}{2}-l&=w\\ \frac{36}{2}-8&=w\\ 18-8&=w\\ 10&=w \end{align*}\)
For a length of \(10\)
\(\begin{align*} \frac{P}{2}-l&=w\\ \frac{36}{2}-10&=w\\ 18-10&=w\\ 8&=w \end{align*}\)
For a length of \(12\)
\(\begin{align*} \frac{P}{2}-l&=w\\ \frac{36}{2}-12&=w\\ 18-12&=w\\ 6&=w \end{align*}\)
For a length of \(15\)
\(\begin{align*} \frac{P}{2}-l&=w\\ \frac{36}{2}-15&=w\\ 18-15&=w\\ 3&=w \end{align*}\)
The area of a parallelogram can be calculated using the formula \(A=bh\), where \(b\) is the length of the base and \(h\) is the height.
Rewrite this formula in terms of the length of the base.
\(\begin{align*} A&=bh\\ \class{timed in1}{\frac{A}{h}}& \; \class{timed in1}{=b} \end{align*}\)
The formula \(V=\pi r^2h\) can be used to determine the volume of a cylinder with radius \(r\) and height \(h\).
\(V\)
\(=\pi r^2h\)
\(=(\pi r^2)h\)
\(\dfrac{V}{\pi r^2}\)
\(=h\)
\(=(\pi h)r^2\)
\(\dfrac{V}{\pi h}\)
\(=r^2\)
\(\sqrt {\dfrac{V}{\pi h}}\)
\(=r \mbox{ , since } r\gt 0\)
To determine the average of three numbers, \(x\), \(y\), and \(z\), add these numbers and divide by \(3\).
The formula \(A=\dfrac{x+y+z}{3}\) can be used to model this situation, where \(A\) is the average.
To solve for \(x\), begin by multiplying by \(3\), then subtract \(y\) and \(z\).
\(A\)
\(=\dfrac{x+y+z}{3}\)
\(3A\)
\(=x+y+z\)
\(3A-y-z\)
\(=x\)
\(\begin{align*} 3A-y-z&=x\\ 3(75)-48-94&=x\\ 225-48-94&=x\\ 83&=x \end{align*}\)
Therefore, the third number is \(83\).
The Pythagorean Theorem, \(a^2+b^2=c^2\), relates the lengths of the three sides of a right-angled triangle, where \(c\) is the hypotenuse. Solve this formula for side \(b\).
Subtract \(a^2\) and then take the square root to isolate \(b\).
\(a^2+b^2\)
\(=c^2\)
\(b^2\)
\(=c^2-a^2\)
\(b\)
\(=\sqrt{c^2-a^2} ,\mbox{ since } b \gt 0\)
The formula for the perimeter of a trapezoid is \(P=a+b+c+d\) and its area is \(A=\dfrac{h(a+b)}{2}\), where \(h\), \(a\), \(b\), \(c\), and \(d\) are shown in the diagram.
Can you isolate \(a\) in both equations?
Rearrange to isolate \(a\).
\(= a + b + c + d\)
\(P - b - c - d\)
\(= a\)
\(=\dfrac{h(a+b)}{2}\)
\(2A\)
\(=h(a+b)\)
\(\dfrac{2A}{h}\)
\(=a + b\)
\(\dfrac{2A}{h} - b\)
\(=a\)