John can walk 4 miles per hour
Answer:
x=4
Step-by-step explanation:
6x+3 = 27
Subtract 3 from each side
6x+3-3=27-3
6x = 24
Divide each side by 6
6x/6 = 24/6
x = 4
a. Factorize the denominator:
![\dfrac{x+14}{x^2-2x-8}=\dfrac{x+14}{(x-4)(x+2)}](https://tex.z-dn.net/?f=%5Cdfrac%7Bx%2B14%7D%7Bx%5E2-2x-8%7D%3D%5Cdfrac%7Bx%2B14%7D%7B%28x-4%29%28x%2B2%29%7D)
Then we're looking for
such that
![\dfrac{x+14}{x^2-2x-8}=\dfrac a{x-4}+\dfrac b{x+2}](https://tex.z-dn.net/?f=%5Cdfrac%7Bx%2B14%7D%7Bx%5E2-2x-8%7D%3D%5Cdfrac%20a%7Bx-4%7D%2B%5Cdfrac%20b%7Bx%2B2%7D)
![\implies x+14=a(x+2)+b(x-4)](https://tex.z-dn.net/?f=%5Cimplies%20x%2B14%3Da%28x%2B2%29%2Bb%28x-4%29)
If
, then
; if
, then
. So we have
![\dfrac{x+14}{x^2-2x-8}=\dfrac3{x-4}-\dfrac2{x+2}](https://tex.z-dn.net/?f=%5Cdfrac%7Bx%2B14%7D%7Bx%5E2-2x-8%7D%3D%5Cdfrac3%7Bx-4%7D-%5Cdfrac2%7Bx%2B2%7D)
as required.
b. Same setup as in (a):
![\dfrac{-3x^2+5x+6}{x^3+x^2}=\dfrac{-3x^2+5x+6}{x^2(x+1)}](https://tex.z-dn.net/?f=%5Cdfrac%7B-3x%5E2%2B5x%2B6%7D%7Bx%5E3%2Bx%5E2%7D%3D%5Cdfrac%7B-3x%5E2%2B5x%2B6%7D%7Bx%5E2%28x%2B1%29%7D)
We want to find
such that
![\dfrac{-3x^2+5x+6}{x^2(x+1)}=\dfrac ax+\dfrac b{x^2}+\dfrac c{x+1}](https://tex.z-dn.net/?f=%5Cdfrac%7B-3x%5E2%2B5x%2B6%7D%7Bx%5E2%28x%2B1%29%7D%3D%5Cdfrac%20ax%2B%5Cdfrac%20b%7Bx%5E2%7D%2B%5Cdfrac%20c%7Bx%2B1%7D)
Quick aside: for the second term, since the denominator has degree 2, we should be looking for another constant
such that the numerator of the second term is
. We always want the polynomial in the numerator to have degree 1 less than the degree of the denominator. But we would end up determining
anyway.
![\implies-3x^2+5x+6=ax(x+1)+b(x+1)+cx^2](https://tex.z-dn.net/?f=%5Cimplies-3x%5E2%2B5x%2B6%3Dax%28x%2B1%29%2Bb%28x%2B1%29%2Bcx%5E2)
If
, then
; if
, then
. Expanding everything on the right then gives
![-3x^2+5x+6=ax^2+ax+bx+b+cx^2=(a-2)x^2+(a+6)x+6](https://tex.z-dn.net/?f=-3x%5E2%2B5x%2B6%3Dax%5E2%2Bax%2Bbx%2Bb%2Bcx%5E2%3D%28a-2%29x%5E2%2B%28a%2B6%29x%2B6)
which tells us
and
; in both cases, we get
. Then
![\dfrac{-3x^2+5x+6}{x^2(x+1)}=-\dfrac1x+\dfrac6{x^2}-\dfrac2{x+1}](https://tex.z-dn.net/?f=%5Cdfrac%7B-3x%5E2%2B5x%2B6%7D%7Bx%5E2%28x%2B1%29%7D%3D-%5Cdfrac1x%2B%5Cdfrac6%7Bx%5E2%7D-%5Cdfrac2%7Bx%2B1%7D)
as required.
Answer: a) 9.23×10⁻⁶ and b) 0.00144
Step-by-step explanation:
Since we have given that
Number of cards = 52
a) Probability that we draw 3 aces and 2 kings.
As we know that
Number of aces = 4
Number of kings = 4
Number of cards drawn = 5
So, the probability becomes,
![\dfrac{^4C_3\times ^4C_2}{^{52}C_5}=\dfrac{24}{2598960}=9.23\times 10^{-6}](https://tex.z-dn.net/?f=%5Cdfrac%7B%5E4C_3%5Ctimes%20%5E4C_2%7D%7B%5E%7B52%7DC_5%7D%3D%5Cdfrac%7B24%7D%7B2598960%7D%3D9.23%5Ctimes%2010%5E%7B-6%7D)
(b) a "full house" (3 cards of one kind, 2 cards of another kind)
Since there are 13 sets of each type and we have to select and 2 kinds of it, so, it becomes ![2\times ^{13}C_2](https://tex.z-dn.net/?f=2%5Ctimes%20%5E%7B13%7DC_2)
So, it becomes,
![\dfrac{2\times ^{13}C_2\times ^4C_3\times ^4C_2}{^{52}C_5}=\dfrac{24\times 156}{2598960}=\dfrac{3744}{2598960}=0.00144](https://tex.z-dn.net/?f=%5Cdfrac%7B2%5Ctimes%20%5E%7B13%7DC_2%5Ctimes%20%5E4C_3%5Ctimes%20%5E4C_2%7D%7B%5E%7B52%7DC_5%7D%3D%5Cdfrac%7B24%5Ctimes%20156%7D%7B2598960%7D%3D%5Cdfrac%7B3744%7D%7B2598960%7D%3D0.00144)
Hence, a) 9.23×10⁻⁶ and b) 0.00144
Answer:
27.72 cm^2
Step-by-step explanation:
Parallelogram's area = base x height
= 3.3 x 8.4
= 27.72 cm^2