Answer:
The numbers are
5.5 and 9.5,
Step-by-step explanation:
If the numbers are x and y:
x + y = 15
x - y = 4
If we add these 2 equations we eliminate y, so
2x = 19
x = 19/2 = 9.5.
Substituting in the first equation for x:
9. 5 + y = 15
y = 15 - 9.5 = 5.5
If the line RS is tangent to the circle, then it is at a 90 degree angle to the circle at the point of contact.
Now that we know 2 angles, we can find the 3rd:
180 - 90 - 32 = 58
It will be the third on because you will be multiplying the same number twice
Answer:
![Rate = 6.7726](https://tex.z-dn.net/?f=Rate%20%3D%206.7726)
Step-by-step explanation:
Given
at ![x =2](https://tex.z-dn.net/?f=x%20%3D2)
Required
The instantaneous rate of change
We have:
![f(x) = x^x](https://tex.z-dn.net/?f=f%28x%29%20%3D%20x%5Ex)
The instantaneous rate of change is:
![\lim_{h \to 0} \frac{f(a + h) -f(a)}{h}](https://tex.z-dn.net/?f=%5Clim_%7Bh%20%5Cto%200%7D%20%5Cfrac%7Bf%28a%20%2B%20h%29%20-f%28a%29%7D%7Bh%7D)
implies that: ![a = 2](https://tex.z-dn.net/?f=a%20%3D%202)
So, we have:
![\frac{f(a + h) -f(a)}{h} = \frac{f(2 + 0.01) -f(2)}{0.01} = \frac{f(2.01) -f(2)}{0.01} = \frac{2.01^{2.01} - 2^2}{0.01} = 6.840403](https://tex.z-dn.net/?f=%5Cfrac%7Bf%28a%20%2B%20h%29%20-f%28a%29%7D%7Bh%7D%20%3D%20%5Cfrac%7Bf%282%20%2B%200.01%29%20-f%282%29%7D%7B0.01%7D%20%3D%20%5Cfrac%7Bf%282.01%29%20-f%282%29%7D%7B0.01%7D%20%3D%20%5Cfrac%7B2.01%5E%7B2.01%7D%20-%202%5E2%7D%7B0.01%7D%20%3D%206.840403)
Keep reducing h but set a constant at 2
![\frac{f(a + h) -f(a)}{h} = \frac{f(2 + 0.001) -f(2)}{0.001} = \frac{f(2.001) -f(2)}{0.001} = \frac{2.001^{2.001} - 2^2}{0.001} = 6.779327](https://tex.z-dn.net/?f=%5Cfrac%7Bf%28a%20%2B%20h%29%20-f%28a%29%7D%7Bh%7D%20%3D%20%5Cfrac%7Bf%282%20%2B%200.001%29%20-f%282%29%7D%7B0.001%7D%20%3D%20%5Cfrac%7Bf%282.001%29%20-f%282%29%7D%7B0.001%7D%20%3D%20%5Cfrac%7B2.001%5E%7B2.001%7D%20-%202%5E2%7D%7B0.001%7D%20%3D%206.779327)
![\frac{f(a + h) -f(a)}{h} = \frac{f(2 + 0.0001) -f(2)}{0.0001} = \frac{f(2.0001) -f(2)}{0.0001} = \frac{2.0001^{2.0001} - 2^2}{0.0001} = 6.773262](https://tex.z-dn.net/?f=%5Cfrac%7Bf%28a%20%2B%20h%29%20-f%28a%29%7D%7Bh%7D%20%3D%20%5Cfrac%7Bf%282%20%2B%200.0001%29%20-f%282%29%7D%7B0.0001%7D%20%3D%20%5Cfrac%7Bf%282.0001%29%20-f%282%29%7D%7B0.0001%7D%20%3D%20%5Cfrac%7B2.0001%5E%7B2.0001%7D%20-%202%5E2%7D%7B0.0001%7D%20%3D%206.773262)
![h = 0.00001](https://tex.z-dn.net/?f=h%20%3D%200.00001)
![\frac{f(a + h) -f(a)}{h} = \frac{f(2 + 0.00001) -f(2)}{0.00001} = \frac{f(2.00001) -f(2)}{0.00001} = \frac{2.00001^{2.00001} - 2^2}{0.00001} = 6.772656](https://tex.z-dn.net/?f=%5Cfrac%7Bf%28a%20%2B%20h%29%20-f%28a%29%7D%7Bh%7D%20%3D%20%5Cfrac%7Bf%282%20%2B%200.00001%29%20-f%282%29%7D%7B0.00001%7D%20%3D%20%5Cfrac%7Bf%282.00001%29%20-f%282%29%7D%7B0.00001%7D%20%3D%20%5Cfrac%7B2.00001%5E%7B2.00001%7D%20-%202%5E2%7D%7B0.00001%7D%20%3D%206.772656)
![h = 0.000001](https://tex.z-dn.net/?f=h%20%3D%200.000001)
![\frac{f(a + h) -f(a)}{h} = \frac{f(2 + 0.000001) -f(2)}{0.000001} = \frac{f(2.000001) -f(2)}{0.000001} = \frac{2.000001^{2.000001} - 2^2}{0.000001} = 6.772595](https://tex.z-dn.net/?f=%5Cfrac%7Bf%28a%20%2B%20h%29%20-f%28a%29%7D%7Bh%7D%20%3D%20%5Cfrac%7Bf%282%20%2B%200.000001%29%20-f%282%29%7D%7B0.000001%7D%20%3D%20%5Cfrac%7Bf%282.000001%29%20-f%282%29%7D%7B0.000001%7D%20%3D%20%5Cfrac%7B2.000001%5E%7B2.000001%7D%20-%202%5E2%7D%7B0.000001%7D%20%3D%206.772595)
![h = 0.0000001](https://tex.z-dn.net/?f=h%20%3D%200.0000001)
![\frac{f(a + h) -f(a)}{h} = \frac{f(2 + 0.0000001) -f(2)}{0.0000001} = \frac{f(2.0000001) -f(2)}{0.0000001} = \frac{2.0000001^{2.0000001} - 2^2}{0.0000001} = 6.772589](https://tex.z-dn.net/?f=%5Cfrac%7Bf%28a%20%2B%20h%29%20-f%28a%29%7D%7Bh%7D%20%3D%20%5Cfrac%7Bf%282%20%2B%200.0000001%29%20-f%282%29%7D%7B0.0000001%7D%20%3D%20%5Cfrac%7Bf%282.0000001%29%20-f%282%29%7D%7B0.0000001%7D%20%3D%20%5Cfrac%7B2.0000001%5E%7B2.0000001%7D%20-%202%5E2%7D%7B0.0000001%7D%20%3D%206.772589)
![h = 0.00000001](https://tex.z-dn.net/?f=h%20%3D%200.00000001)
![\frac{f(a + h) -f(a)}{h} = \frac{f(2 + 0.00000001) -f(2)}{0.00000001} = \frac{f(2.00000001) -f(2)}{0.00000001} = \frac{2.00000001^{2.00000001} - 2^2}{0.00000001} = 6.772589](https://tex.z-dn.net/?f=%5Cfrac%7Bf%28a%20%2B%20h%29%20-f%28a%29%7D%7Bh%7D%20%3D%20%5Cfrac%7Bf%282%20%2B%200.00000001%29%20-f%282%29%7D%7B0.00000001%7D%20%3D%20%5Cfrac%7Bf%282.00000001%29%20-f%282%29%7D%7B0.00000001%7D%20%3D%20%5Cfrac%7B2.00000001%5E%7B2.00000001%7D%20-%202%5E2%7D%7B0.00000001%7D%20%3D%206.772589)
Notice that:
for
and ![h = 0.0000001](https://tex.z-dn.net/?f=h%20%3D%200.0000001)
Hence, the instantaneous rate of change is:
![Rate = 6.772589](https://tex.z-dn.net/?f=Rate%20%3D%206.772589)
---- approximated
Answer:
![\frac{17x-2}{5x-2}](https://tex.z-dn.net/?f=%5Cfrac%7B17x-2%7D%7B5x-2%7D)
Step-by-step explanation:
10x/5x-2 + 7x-2/5x-2
![\frac{10x}{5x-2} + \frac{7x-2}{5x-2}](https://tex.z-dn.net/?f=%5Cfrac%7B10x%7D%7B5x-2%7D%20%2B%20%5Cfrac%7B7x-2%7D%7B5x-2%7D)
Here both fractions have same denominator 5x-2
When the denominators are same then we can combine the numerators
we put denominator only once
![\frac{10x+7x-2}{5x-2}](https://tex.z-dn.net/?f=%5Cfrac%7B10x%2B7x-2%7D%7B5x-2%7D)
Now we simplify it
Combine like terms: 10x+7x becomes 17x
![\frac{17x-2}{5x-2}](https://tex.z-dn.net/?f=%5Cfrac%7B17x-2%7D%7B5x-2%7D)
There is no common factor for the top and bottom
So we cannot simplify further