Lesson 2 · from §111 of the 1910 text
Radicals in Formulas and Problems
Radicals in formulas
Many formulas of geometry and physics contain radicals or squares. To solve such a formula for a letter, isolate the radical (or the square), then raise both members to the proper power, exactly as in the rule above.
Example. Solve t = \pi\sqrt{\dfrac lg} for l.
Dividing by \pi, \dfrac t\pi = \sqrt{\dfrac lg}. Squaring, \dfrac{t^2}{\pi^2} = \dfrac lg. Therefore l = \dfrac{gt^2}{\pi^2}.
When the letter sought appears squared (as R in A = \pi R^2), extracting the square root gives two values with the sign \pm: R = \pm\sqrt{\dfrac A\pi}. In a problem only the positive value has a meaning, but the equation has both roots.
Typing answers. Letters stand for positive numbers. Type a double root with ± or +-: ±sqrt(A/pi), or list both: sqrt(A/pi), -sqrt(A/pi). Type \pi as pi.
Numerical problems. Use g = 32 feet per second per second and 1 mile = 5280 feet.
Quick check
Solve for r: S = 4\pi r^2.
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Solve each formula for the letter named:
Solve for v: E = \dfrac{mv^2}{2}.
Solve for l and for g: t = \pi\sqrt{\dfrac lg}.
Solve for R: A = \pi R^2.
Solve for R: V = \dfrac{\pi R^2H}{3}.
Solve for R: \dfrac{R^2}{4} + S^2 = R^2.
Solve for R: \sqrt{R^2 - \left(\dfrac{R\sqrt3}{2}\right)^2} = A.
Solve for R and for a: \left(\dfrac R2\sqrt{2 - \sqrt2}\right)^2 = R^2 - a^2.
If a bullet is fired vertically, the least velocity, V, which it may have so that it will never return to the earth is given by the equation V = \sqrt{2gR} (g = 32 feet per second, R = 4000 miles). Find the velocity in miles per second to the nearest whole number.
The greatest distance, x, that a ball can be thrown with velocity v (in feet per second) across a level field is given by one root of the equation .976v^2x - gx^2 = 0 (g = 32 feet). Under the conditions just stated a ball is thrown with a velocity of 100 feet per second. How far from the thrower did it strike the ground?
The greatest distance a baseball has been thrown is about 400 feet. With what velocity did it leave the thrower's hand? (Use the equation of problem 42.)