A cylindrically shaped piece of collagen (a substance found in the body in connective tissue) is being stretched by a force that increases from $0 \text { to } 3.0 \times 10 ^ { - 2 } \mathrm { N }.$ The length and radius of the collagen are, respectively, 2.5 and 0.091 cm, and Young's modulus is $3.1 \times 10 ^ { 6 } \mathrm { N } / \mathrm { m } ^ { 2 }$

If the stretching obeys Hooke's law, what is the spring constant k for collagen?

Fill in the blanks with an appropriate word, phrase, or symbol(s). A topological object that resembles a bottle but has only one side is a(n) _ bottle.

Use the function to evaluate the indicated expressions and simplify.

$$f ( x ) = x ^ { 2 } + 1 ; \quad f ( x + 2 ) , f ( x ) + f ( 2 )$$

Yo prefiero ____ la salud y comer bien (Down)

A copper rod

$$(length = 2.0 \mathrm { m } , \text { radius } = 3.0 \times 10 ^ { - 3 } \mathrm { m } )$$

hangs down from the ceiling. A 9.0-kg object is attached to the lower end of the rod. The rod acts as a "spring ," and the object oscillates vertically with a small amplitude. Ignoring the rod's mass, find the frequency f of the simple harmonic motion.

Solve by mental math.

$$\frac { 1 } { 6 } = \frac { a } { 72 }$$

Prove the identity. 2 csc 2x = csc² x tan x

The problem is motivated by the infinite sum

$$\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+....$$

You are going to show that the value of this sum is between 1 and 2 with the help of Exercise 22.4(d). This sum is known as $\zeta(2)$ (where $\zeta$ is the Greek letter zeta and stands for the Riemann zeta function). For your first step, please prove (by induction) that the following inequality holds for all positive integers n:

$$\frac{1}{1^2}+\frac{1}{2^2}+...+\frac{1}{n^2}\gt\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+...+\frac{1}{n(n+1)}\:\:\:\:\:(\ast)$$

For the second step, prove (by induction) this variation of ($\ast$) is true for all positive integers n:

$$\frac{1}{1^2}+\frac{1}{2^2}+...+\frac{1}{n^2}\lt1+\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+...+\frac{1}{n(n+1)}\:\:\:\:\:(\ast \ast)$$

Finally, use ($\ast$), ($\ast \ast$), and Exercise 22.4 (d) to show that $1\leq \zeta(2)\leq 2$.

Write an algebraic expression for: 4 times the sum of x and $\frac{1}{2}$

You hang a 5 kg mass from the end of a rod that is 1.7 m by 2 mm by 2 mm. You observe that the rod’s length increases to 1.74 m. (a) What is the strain?

A solid brass sphere is subjected to a pressure of

$$1.0 \times 10 ^ { 5 } \mathrm { Pa }$$

due to the earth's atmosphere. On Venus the pressure due to the atmosphere is

$$9.0 \times 10 ^ { 6 } \mathrm { Pa }$$

By what fraction

$$\Delta r / r _ { 0 }$$

(including the algebraic sign) does the radius of the sphere change when it is exposed to the Venusian atmosphere? Assume that the change in radius is very small relative to the initial radius.

Find each sum or difference.

$$8.006-6.38$$

Jeff and Stephen go to the mall. The two boys buy a total of 15 T-shirts. Stephen gets three less than twice as many T-shirts as Jeff. a) Write an equation to represent the information in the second sentence. b) Write an equation to represent the information in the third sentence. c) Solve the linear system by substitution to find the number of T-shirts each boy bought. d) If the T-shirts cost $8.99 each, how much did each boy spend before taxes?

Find the critical value

$$t _ { c }$$

for the level of confidence c and sample size n. c = 0.95, n = 24