Very nice to have an accompanied image!Illumination is proportional to the intensity of the source, inversely proportional to the distance squared, and to the sine of angle alpha.so that we can writeI(h)=K*sin(alpha)/s^2 ................(0)where K is a constant proportional to the light source, and a function of other factors.
Also, radius of the table is 4'/2=2', therefore, using Pythagoras theorem,s^2=h^2+2^2 ...........(1), and consequently,sin(alpha)=h/s=h/sqrt(h^2+2^2)..............(2)
Substitute (1) and (2) in (0), we can writeI(h)=K*(h/sqrt(h^2+4))/(h^2+4)=Kh/(h^2+4)^(3/2)
To get a maximum value of I, we equate the derivative of I (wrt alpha) to 0, orI'(h)=0or, after a few algebraic manipulations, I'(h)=K/(h^2+4)^(3/2)-(3*h^2*K)/(h^2+4)^(5/2)=K*sqrt(h^2+4)(2h^2-4)/(h^2+4)^3We see that I'(h)=0 if 2h^2-4=0, giving h=sqrt(4/2)=sqrt(2) feet above the table.
We know that I(h) is a minimum if h=0 (flat on the table) or h=infinity (very, very far away), so instinctively h=sqrt(2) must be a maximum.Mathematically, we can derive I'(h) to get I"(h) and check that I"(sqrt(2)) is negative (for a maximum). If you wish, you could go ahead and find that I"(h)=(sqrt(h^2+4)*(6*h^3-36*h))/(h^2+4)^4, and find that the numerator equals -83.1K which is negative (denominator is always positive).
An alternative to showing that it is a maximum is to check the value of I(h) in the vicinity of h=sqrt(2), say I(sqrt(2) +/- 0.01)we findI(sqrt(2)-0.01)=0.0962218KI(sqrt(2)) =0.0962250K (maximum)I(sqrt(2)+0.01)=0.0962218KIt is not mathematically rigorous, but it is reassuring, without all the tedious work.
Find 29.12% of 6,760,950. You can do this by converting 29.12% to a decimal by moving the decimal 2 places to the left and getting rid of the percent sign. Do this and you figure out 29.12% is .2912 as a decimal. Now you multiply this decimal with 6,760,950 to find what the government takes for taxes from his annual salary. You get 1,968,788.64. This is not your final answer because you want to find the amount of money he keeps after taxes. So you'd subtract 1,968,788.64 from 6,760,950. This gives you 4,792,161.36. This is his take-home pay, but you have to round to the nearest yen, so your final answer is 4,792,161.
Answer:
The flag pole is 38.96 feet tall.
Step-by-step explanation:
The angle of depression from the top of the flagpole to an object on the ground is 33°.
The object is 60 feet from the base of the cliff.
We need to find the height of the flag pole.
Here, base, B = 60 feet
We know that,
Put all the values to find P.
So, the flag pole is 38.96 feet tall.
First bring all the x's to one side.
then subtract them.
then divide 24 by 3 to get the x alone and get your answer.
11x-8x=24
3x=24
x=8
Answer:
x=13
Step-by-step explanation:
ok so let x+y=22
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x and y are the 2 unkown Number
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x-y=4
their difference is 4
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add both equations
(x+y=22)
+(x-y=4)
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y's cancel out
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2x=26
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x=13
