<span>Question No.2
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speed= distance/ time
1200km/0.006s= 200 000 km/s
<span>Question 4
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wavelength = wavespeed / frequency
wavespeed = 16/8 = 2
Question 5
(400/60s) = 6.6667 waves/ sec (Hz).
<span>Velocity = 0.008m/sec. </span>
<span>Wavelength = (v/f) = 0.008/6.6667 = 0.0012 metre (1.2mm).
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Answer:
12 wooden rulers
23 plastic rulers
Step-by-step explanation:
Total rulers sold - 35
Total profit - $82
1 wooden ruler - $3
1 plastic ruler - $2
$3 x 12 = $36
$2 x 23 = $46
$46 + $36 = $82
12 + 23 = 35 rulers
To prove this inequality we need to consider three cases. We need to see that the equation is symmetric and that switching the variables x and y does not change the equation.
Case 1: x >= 1, y >= 1
It is obvious that
x^y >= 1, y^x >= 1
x^y + y^x >= 2 > 1
x^y + y^x > 1
Case 2: x >= 1, 0 < y < 1
Considering the following sub-cases:
- x = 1, x^y = 1
- x > 1,
Let x = 1 + n, where n > 0
x^y = (1 + n)^y = f_n(y)
By Taylor Expansion of f_e(y) around y = 0,
x^y = f_n(0) + f_n'(0)/1!*y + f_n''(0)/2!*y^2 + ...
= 1 + ln(1 + n)/1!*y + ln(1 + n)^2/2!*y^2 + ...
Since ln(1 + n) > 0,
x^y > 1
Thus, we can say that x^y >= 1, and since y^x > 0.
x^y + y^x > 1
By symmetry, 0 < x < 1, y >= 1, also yields the same.
Case 3: 0 < x, y < 1
We can prove this case by fixing one variable at a time and by invoking symmetry to prove the relation.
Fixing the variable y, we can set the expression as a function,
f(x) = x^y + y^x
f'(x) = y*x^(y-1) + y^x*ln y
For all x > 0 and y > 0, it is obvious that
f'(x) > 0.
Hence, the function f(x) is increasing and hence the function f(x) would be at its minimum when x -> 0+ (this means close to zero but always greater than zero).
lim x->0+ f(x) = 0^y + y^0 = 0 + 1 = 1
Thus, this tells us that
f(x) > 1.
Fixing variable y, by symmetry also yields the same result: f(x) > 1.
Hence, when x and y are varying, f(x) > 1 must also hold true.
Thus, x^y + y^x > 1.
We have exhausted all the possible cases and shown that the relation holds true for all cases. Therefore,
<span> x^y + y^x > 1
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I have to give credit to my colleague, Mikhael Glen Lataza for the wonderful solution.
I hope it has come to your help.
</span>
Answer:
Step-by-step explanation:
9(c+7)=27
9c+63=27 subtract 63 from both sides
9c= -36 divide by 9
c= -4
3(2r-8)+4r=76 foil
6r-24+4r=76 combine like terms
10r-24=76 add 24 to both sides
10r+100 divide by 10
r= 10
