All Answers 358490 Answers

Sun 25/10/15

1)  Trig manipulation question       Thanks Omi67 and Melody.

      http://web2.0calc.com/questions/trig-help_4130

2) Maximizing a function given a system of contraints           Melody

    http://web2.0calc.com/questions/linear-programming-pls-help#r1

3) I got a bit carried away with this differentiation.  ://      Thanks also to Omi67

    http://web2.0calc.com/questions/differentiation_1#r1

4) Trig, find the height of the helicopter.    Thanks Alan.

     http://web2.0calc.com/questions/trigo-sum

Find sin θ if cot θ = - 2 and cos θ < 0.

Thanks Omi for that great answer, 

I have yet another method.

Since cos is negative the answer must be in the 2nd or third quad

Since cot is neg, tan is also neg so it is in the 2nd or  4th quad.

Therefore it is in the 2nd quad where sine is positive!

I am going to put that knowledge aside for a bit and work out the first quadrant equivalent.

\(cot\theta=\frac{2}{1}\\ so\\ tan\theta=\frac{1}{2}\)

Draw a right angled triangle and mark one angle as theta, then put

1 on the opposite side and

2 on the adjacent side so that tan theta = 1/2

Find the hypotenuse using pythagoras   

\(hypotenuse=\sqrt{1^2+2^2}=\sqrt{5}\)

\(sin\theta = \frac{opp}{hyp}=\frac{1}{\sqrt5}=+\frac{\sqrt5}{5}\)

(positive for our final answer too because theta is in  the 2nd quad where sine is positive.)

At this point is is FINISHED.  You are not asked to find theta :))

so

These two attitudes should not be bundles together!

OK you find maths is hard and you most likely dislike it.

That is ok, there are lots of things I dislike too, just maths is not one of them.

If life suxs for you that has nothing to do with maths.  If you think it does then it is your thoughts that are the problem, not the mathematics.

------------

A suggestion for you.  If you want to do better at maths and do not want it to totally stress you out then put time into practicing and fully understanding the easier stuff.  Do not get too stressed about things that are too difficult.  Maybe you will even start not disliking it too much :)

For people who excel at school and want to have highly professional careers, school results are very important.  For other people a few marks more or less makes little difference.  Do the best you reasonably can but don't let it make you miserable!  There is MUCH more to life than school.

The ladder represents the hypotenuse. The distance from the ground is one of the legs.

So plug in the varaibles to the pythagorean theorem; a² + b² = c²

36² + b² = 39²

(simplified) 1296 + b² = 1521

Cancel out the 1296 by subtracting both sides, and then solve for b² by taking the square root of both sides.

b = √225 = 15

And there you go. You now know the ladder must be placed 15 feet from the house. Hope this was helpful!

It is a "mixed fraction"!!!!!.

A helium-neon laser emits light with a wavelength of 6.33x10-7 m. What is the frequency of this light if it is traveling the speed of light?

Select one:

a. 4.19x10-40 Hz

b. 4.74x1014 Hz

c. 189 Hz

d. 4.74 Hz

Use this formula:Wavelength=c/v, where c is the speed of light, and v the frequency.

6.33 X 10^-7=3 X 10^8/v

v=3 X 10^8/6.33 X 10^-7

v=4.74 X 10^14 Hertz, so obviously your answer is "b"

Hi Omi

I agree, but you forgot the log₈(x) term.  

ln(x-1)/ln(8) - ln(x)/ln(8) = 1 

See Guest's development in #4 above.

Hallo Alan, I think in such a way

???

Solve for x:
(log(x-1))/(log(8))-(log(x))/(log(8)) = 1

Rewrite the left hand side by combining fractions. (log(x-1))/(log(8))-(log(x))/(log(8))  =  (log(x-1)-log(x))/(log(8)):
(log(x-1)-log(x))/(log(8)) = 1

Multiply both sides by log(8):
log(x-1)-log(x) = log(8)

log(x-1)-log(x) = log(x-1)+log(1/x) = log((x-1)/x):
log((x-1)/x) = log(8)

Cancel logarithms by taking exp of both sides:
(x-1)/x = 8

Multiply both sides by x:
x-1 = 8 x

Subtract 8 x-1 from both sides:
-7 x = 1

Divide both sides by -7:
Answer: | 
| x = -1/7(assuming a complex-valued Logarithm)

9 + 10 equals 19.

\((a^{-1}+b^{-1})(a^{-1}-b^{-1})^3\\ =\left(\frac{1}{a}+\frac{1}{b}\right)\times\left(\frac{1}{a}-\frac{1}{b}\right)^3\\ =\left(\frac{b}{ab}+\frac{a}{ab}\right)\times\left(\frac{b}{ab}-\frac{a}{ab}\right)^3\\ =\left(\frac{b+a}{ab}\right)\times\left(\frac{b-a}{ab}\right)^3\\ =\frac{(b+a)(b-a)^3}{a^4b^4}\\ \)

You may finish here but you may want to expand it.  

\(=\frac{(b+a)(b-a)\times(b-a)^2}{a^4b^4}\\ =\frac{(b^2-a^2)\times(b^2-2ab+a^2)}{a^4b^4}\\ =\frac{b^4-2ab^3+2a^3b-a^4}{a^4b^4}\\ \)

Hmmm

\(\log_8{(x-1)}-\log_8{x}=1 \rightarrow\log_8{(\frac{x-1}{x})}=1\rightarrow\frac{x-1}{x}=8^1\rightarrow\frac{x-1}{x}=8\\\\ \rightarrow x-1=8x\rightarrow7x=-1\rightarrow x=-\frac{1}{7}\)

Slightly odd, as the original individual logarithms would only have complex values, though the combined log would be real (the imaginary parts cancel out).

Hallo guest

simplify:(a-^1+b^-1)(a^-1-b^-1)^3

-((a-b)^3 (a+b)^3)/(a^6 b^6), or

-(a^2-b^2)^3/(a^6 b^6), or

1/a^6-3/(a^4 b^2)+3/(a^2 b^4)-1/b^6

First multiply it by 100 to get the percent
.000000025 x 100 = .0000025 %     or   25 ten millionths of a per cent

I have made a small mistake.

greatest common factor of 10 and 4,

4=2 X 2

10=2 X 5

As you can see, 2 is the GCF.

Hallo guest

An electromagnet wave has a frequency of 2.5 x 1012 Hz and a speed of 3 x 108 m/s. Use the equation for a wavelength to calculate the wavelength of this wave.

Wavelength=c/v, where c is the speed of light, and v is the frequency.

W=3 X 10^8/2.5 X 10^12

W=0.00012m=1.2 X 10^-4=.12Km

As you can see the answer is obviously "B"

what is .000000025 expressed as a percentage? Is it 25 quadrillion of a percent?

NO!.

As a percentage it is=2.5 X 10^-6%

It is actually 1 / 40,000,000 of 1%, or one forty millionth of 1%.

sin(theta) = -sin(cot^(-1)(2)-pi c1)

log(8, x-1)-log(8, x) = 1

(log(x-1))/(log(8))-(log(x))/(log(8)) = 1

2777777777777777777777777777777777335333333333333333333333333333333333457=

999999 9999999999 9999999999 9999999841 x 2777777 7777777777

7777777777 7777777777.

A scientist collects a 20­g sample of radioactive Iodine­131. After a time,

only 5 g of the sample is radioactive. If the half­life of Iodine­131 is 8.07

days, how long did the scientist allow the sample to decay?

Go to this link and it is all explained there:

http://www.mathsisfun.com/data/standard-deviation.html