Question

Question #1)

Jeremy determines that √9= 9^1/2. Part of his work is shown

√9=3=3^1=3^1/2 + ^1/2=____=9^1/2

Which expression or equation should be placed in the blank to correctly complete Jeremy's work?

A: (3^2)^1
B: 3^1/2 + 3^1/2
C: 3^1/2 x 3^1/2= (3x3)^1/2
D: 3^1/2 x 3^1/2= (3x3)^1/2+^1/2



Question #2)

A bird drops a stick from the top of Miami Tower. The height of the stick after x seconds is given by f(x) = 625-16x^2.

Select all the correct interpretations of the coordinates of the point at the maximum of the function f(x).

A: The time it takes the stick to hit the ground
B: The time when the stick is at its highest point
C: The height of the stick when it is dropped from Miami Tower
D: The time when the stick is dropped from the top of Miami Tower
E: The distance of the stick from Miami Tower when it hits the ground.

Answer 1

Answer with explanation:

1.

The Identity Wrote by Jeremy

`√(9)=9^{(1)/(2)}`

To prove this identity , his work is as follows

`√(9)=3\\\\=3^1\\\\=3^{ (1)/(2)+(1)/(2)}\\\\=3^{ (1)/(2)} * 3^{ (1)/(2)}\\\\=(3* 3)^{ (1)/(2)}\\\\=9^{ (1)/(2)}`

Used the following law of indices

`1.\rightarrow x^(a+b)=x^a* x^b\\\\2.\rightarrow x^a * y^a=(x y)^a`

Option C

`\rightarrow 3^{ (1)/(2)} * 3^{ (1)/(2)}= (3* 3)^{ (1)/(2)}`

2.

Height of stick after x seconds is given by

f(x)=625 -16 x²

Replacing , f(x) by , y in the above equation

y= 625 -16 x²

16 x²= - (y-625)

→when x=0, y=625

Coordinates of vertex = (0, 625)

⇒Maximum of the function is given by , when time , x=0,

Maximum height = 625

Following are the correct interpretations of the coordinates of the point at the maximum of the function f(x).

B: The time when the stick is at its highest point.

C: The height of the stick when it is dropped from Miami Tower.

E: The distance of the stick from Miami Tower when it hits the ground.

Answer 2

1. From Jeremy's work:
√9
=3^(1/2)×3^(1/2)
multiplying number with the same powers is the same as multiplying the number first then raising it to the power.
=(3×3)^(1/2)
from above we conclude that the answer is:
C: 3^1/2 x 3^1/2= (3x3)^1/2

2. Given that the height of the stick at time t seconds after it was dropped is given by the function:
f(x)=625-16x²
At the maximum point f'(x)=0
thus
f'(x)=-32x=0
thus at maximum point x=0, y=625. The coordinates is (0,625), this represents the distance of the stick from Miami Tower when it hits the ground.

Answer: E