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.