Answer:
∅ = 30°
Step-by-step explanation:
given data
side 1 = 6 inches
side 2 = 
inches
to find out
angle opposite the leg that is 6 inches long
solution
we will use pythagoras theorem that is express as
hypotenuse² = perpendicular² + base² .......................1
put here value and we get hypotenuse
hypotenuse² = 6² + ( )²
solve it we get
hypotenuse² = 144
hypotenuse = 12 inches
now we apply here sin∅ that is
sin∅ = 
................2
put here value and we get
put here value and we get
sin∅ = 
∅ = 
solve it we get
∅ = 30°
A
to calculate the slope m , use the ' gradient formula
m = (y₂ - y₁ ) / (x₂ - x₁ )
with (x₁, y₁ ) = (1, 3) and (x₂, y₂ ) = (5, 4)
m = 
= 
→ A
Note that
,
,
, and so on. The function
is continuous and increasing for all
, so when
, we have
; when
,
; and so on.
This means
which means we can capture the number of digits of
with the function
.
So the problem is the same as finding positive integer solutions to
We know that
has one digit, so clearly this must be a solution. We need to show that this is the only solution.
Recall that
![\dfrac{\mathrm d}{\mathrm dx}[\log_{10}x+1]=\dfrac1{\ln10\,x}](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5B%5Clog_%7B10%7Dx%2B1%5D%3D%5Cdfrac1%7B%5Cln10%5C%2Cx%7D)
, while
![\dfrac{\mathrm d}{\mathrm dx}[x]=1](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5Bx%5D%3D1)
. This means
increases at a much slower rate than
as
. We know the two functions intersect when
. Therefore it's clear that
for all
.
Now, it's always the case that
, so we're essentially done:
which means there are no other solutions than
.
The slop is stocks falling you dont need it
