Download 5.3 Challenge: Skills and Applications

LESSON
NAME _________________________________________________________ DATE ___________
5.3
Challenge: Skills and Applications
For use with pages 264–270
1. Einstein’s Special Theory of Relativity states that an object’s mass m
increases with its velocity v. This increase is not noticeable until the v is
a significant fraction of the speed of light c. The mass m of the object is
given by the formula
m
m0
,
v2
1 2
c
where m0 is the mass of the object when it is at rest.
a. Suppose an object’s velocity is
1
4
the speed of light. Express the mass
of the object in terms of its rest mass, in simplest form.
b. Find the velocity (in terms of c) at which an object’s mass is twice its
rest mass.
2. For numbers x close to 0, the approximation
1 x 1 x
2
is fairly accurate.
a. Use a calculator to find the actual value and the approximate value for
x 0.12. What is the difference between them?
b. A better approximation is given by
1 x 1 x2
x
.
2
8
By how much does this estimate differ from the actual value, for the
value of x in part (a)?
3. You probably know that in general a2 b2 a b. Show that if the
Lesson 5.3
two sides of this inequality are equal, then at least one of a or b is 0.
(Hint: Square both sides.)
4. a. Find, in terms of x, the length of the hypotenuse of a right triangle
whose legs have length x and 4x, where x is a positive number. Repeat
for legs of length x and 7x. Simplify your answers.
b. Based on your answers to part (a), make a conjecture about the length
of the hypotenuse of a right triangle with legs x and nx, where n is a
positive integer. Show by calculation that your conjecture is correct.
5. a. Solve the equation a cx d for x, in terms of a, c, and d.
Assume a ≥ 0.
2
b. Use the formula you found in part (a) to express, in simplest form,
the value of x when a 50, c 12, and d 18.
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Algebra 2
Chapter 5 Resource Book
49