Math problems...trigonometry and various topics...?

Question

1. Write 3 log (x+2) + log 8x - 2 log (x+8) as a single logarithm.
(all of them to the base 2)

2. A certain kind of bacteria doubles every three days. If initially there are 100 bacteria, about how many will there be after 2 days?

3. What is the range of y= 3^x + 4 ?

I don't really know how to put down the next problems, because they have a symbol in them and I don't know what it is (our class hasn't reached this far yet). Google isn't helping much either. The symbol that I'm talking about looks like a zero with a horizontal bar at the middle. I'm just going to denote its presence by (symbol).

4. If sin (symbol) = 3/8 in Q1, what is cos (symbol) ?

5. Write in terms of sine and cosine (1-tan^2 (symbol) ) / (1+tan^2 (symbol) ).

6. If 0 greater than or equal to (symbol) less than or equal to 360 degrees, solve for (symbol) in 2cos^2(symbol) = 1 + sin (symbol).

Thanks very much! Solutions/Explanations would be much appreciated.

Answer 1

1. Write 3 log (x+2) + log 8x - 2 log (x+8) as a single logarithm.
(all of them to the base 2).

I assume all the logs are base 2. Then

3 log (x+2) + log 8x - 2 log (x+8)
= log (x+2)³ + log 8x - log (x+8)²
= log {(x+2)³ (8x) / (x+8)²}
= log {(8x) (x+2)³ / (x+8)²}

2. A certain kind of bacteria doubles every three days. If initially there are 100 bacteria, about how many will there be after 2 days?

N = number of bacteria after 2 days.

N = 100 * 2^(2/3) = 158.74 or about 159 bacteria

3. What is the range of y= 3^x + 4 ?

The range is the values of x the equation can assume. The range is unrestricted so it is (-∞,∞).

4. If sin θ = 3/8 in Q1, what is cos θ ?

Remember the identity sin² θ + cos² θ = 1. So

cos θ = √(1 - sin² θ) = √{1 - (3/8)²} = √(1 - 9/64)
= √(55/64) = (√55)/8

5. Write in terms of sine and cosine (1-tan² θ) / (1+tan² θ).

(1-tan² θ) / (1+tan² θ) = (1 - sin² θ/cos² θ) / (1 + sin² θ/cos² θ)
= (cos² θ - sin² θ) / (cos² θ + sin² θ) = (cos² θ - sin² θ) / 1
= (cos² θ - sin² θ) = cos 2θ

6. If 0 greater than or equal to θ less than or equal to 360 degrees, solve for θ in 2cos² θ = 1 + sin θ.

This question is a little garbled. But I think you mean to solve for θ on the interval [0°,360°]

2cos² θ = 1 + sin θ
2 - 2sin² θ = 1 + sin θ
2sin² θ + sin θ - 1 = 0
(2sin θ - 1)(sin θ + 1) = 0
sin θ = 1/2, -1
θ = 30°, 150°, 270°

Answer 2

Math problems...trigonometry and various topics...?
1. Write 3 log (x+2) + log 8x - 2 log (x+8) as a single logarithm.
(all of them to the base 2)
log8x(x+2)^3/(x+8)^2

2. A certain kind of bacteria doubles every three days. If initially there are 100 bacteria, about how many will there be after 2 days?
A=A0e^kt
A0=100
2=e^3k as it doubles every 3 days
solve for k
theplugand chug

3. What is the range of y= 3^x + 4 ?
all real x

I don't really know how to put down the next problems, because they have a symbol in them and I don't know what it is (our class hasn't reached this far yet). Google isn't helping much either. The symbol that I'm talking about looks like a zero with a horizontal bar at the middle. I'm just going to denote its presence by (symbol).

4. If sin (symbol) = 3/8 in Q1, what is cos (symbol) ?it is theta,a greek letter
adjacent side=rt(64-9)=rt55
so cos theta=rt55/8

5. Write in terms of sine and cosine (1-tan^2 (symbol) ) / (1+tan^2 (symbol) ).
1-tan^2theta/1+tan^2theta=cos 2 theta
in terms of sin and cos
=[1-sin^2theta/cos^2thetsa]/[1+sin^2theta/cos^2theta]
=cos^2theta-sin^2theta

6. If 0 greater than or equal to (symbol) less than or equal to 360 degrees, solve for (symbol) in 2cos^2(symbol) = 1 + sin (symbol
2cos^2theta-1-sintheta=0
2(1-sin^2theta)-1-sintheta=0
2-2sin^2theta-1-sintheta=0
1-2sin^2theta-sintheta=0
use the quadratic formula to solve

looks like the last q is incomplete

Answer 3

Two sides of a triangle have lengths of 16 and 18. The length of third side cannot be greater than the sum 16 + 18 = 34 or less than the difference 18 -- 16 = 2 so answer is 1. D. 1 A store buys pens at 8 for 25 cents and sells them at 2 for 15 cents. To make a profit of $3.50, the store must sell 350 / [15/2 -- 25/8] = 350 / [35/8] = 80 C. 80 pens The second hand of a clock is 5 cm long. In one hour, the tip of the second hand travels a distance of 60*2(pi)*5 cm = 600 pi cm D. 600pi cm Sorry for correction as I mistook second for minute. A man has a 10m X 10m square garden. In the center is a 2m X 2m square patch which he cannot use. He divides his usable space into four congruent rectangular patches, each of which measures (1/4)[10x10 -- 2x2] m^2 = 24 m^2 = 6mx4m as 8mx3m is not possible C. 4m X 6m A wise old owl climbed up a tree. Whose height was exactly ninty plus three. Every day the owl went up eighteen. Every night the owl came down thirteen. If the owl did not pause or stop, its claws first reached the top on [93 -- 18] / [18 -- 13] + 1 day = 75/5 + 1 day = 15 + 1 day = 16th day A. day 16

Answer 4

range means all possible ouytcomes for y...

it will come integers as 3 raised to an integer is an integer...
adding 4 to it wont change
next time give lesser questions

Answer 5

The circle with a line through it is a theta. It means an angle.