Answer to Question #269613 in Math for Jas

Question

The velocity of a tsunami in feet per period

6

9

12

15

18

21

24

Height of tsunami in feet

4

9

16

25

36

49

64

Use the data in the above in inductive reasoning to answer each of the following questions

1. If a pendulum has a length of 49 units what is its period?

2. If the length of a pendulum is quadrupled what happens to its period?

Verify that each of the following statements is a false statement by finding a counterexample

1. X/y=1

2. X+y/ 3= x+1

3. 3 √x2+16= x+4

Answer 1

1.1.

$$2=\dfrac{12}{6}=>\dfrac{16}{4}=4$$

$$2=\dfrac{18}{9}=>\dfrac{36}{9}=4$$

$$2=\dfrac{24}{12}=>\dfrac{64}{16}=4$$

Height of a tsunami increases by 4 times when the velocity is doubled. 

1.2.

$$2=\dfrac{30}{15}=>\dfrac{h}{25}=4=>h=100$$

The height of a tsunami should be 100 ft if its velocity is 30 feet per second.

2. The period of a simple pendulum $T$ is 

$$T=2\pi \sqrt{L/g},$$

where $L$ is the length of the string and $g$ is the acceleration due to gravity.

2.1. Given $L=49\ m, g=9.8\ m/s^2$

$$T=2\pi\sqrt{49\ m/(9.8\ m/s^2)}=2\sqrt{5}\pi \ s\approx14.050\ s$$

2.2.

$$T=2\pi \sqrt{L/g}$$

If you quadruple the length of the string,  the period doubles.

3.1.

Let $x=2, y=1.$ Then

$$x/y=2/1=2\not=1$$

The given statement is False.

3.2.

Let $x=1, y=6.$ Then

$$x+y/3=1+6/3=3$$

$$x+1=1+1=2$$

$$x+y/3=3\not=2=x+1$$

The given statement is False.

3.3.

Let $x=0.$ Then

$$3\sqrt{x^2+16}=3\sqrt{(0)^2+16}=12$$

$$x+4=0+4=4$$

$$3\sqrt{x^2+16}=12\not=4=x+4$$

The given statement is False.