If this question is asked in 5th grade, its answer should be explained at that level.
There are 10 teams to choose for 1st place, having done that there are 9 teams that could be in 2nd place, and that done there remain 8 teams that could be in 3rd place. The total number of choices is...
The only tricky parts are finding the reference angles for
2π/3 and 3π/4
integrand is (siny/cosy)2sinycosydy=2sin2ydy
at x=0 y=0 and at x=1/2 y=π/4
∫2sin2ydy from 0 to π/4
Original fraction 5d/d. Adjusted fraction (5d+12)/(d+12)=2
5d+12=2d+24, 3d=12, d=4
20/4 original fraction, 32/16 when 12 added to both numerator and denominator
a. 4C2/52C2=1/221 two aces out of four/choose 2 cards from deck
b. 48C2/52C2=188/221 two cards not aces/choose 2 cards from deck
5z-20/6z-24=5(z-4)/6(z-4)=5/6 if z≠4 and may be defined as 5/6 by limit if z=4.
Distance= Rate times time and so time=Distance/Rate
time=85/√10,800 if you like rational denominators
x=3±5=8 or -2
7.5/42=10.9/e where e is the length of the elephant's shadow
The ratio test requires that we look at the ratio of the n+1st term to the nth term
((n+1)/n)5→1 as n→∞ and for the second term
5n^4-(n+1)^4=5-(4n^3+6n^2+4n+1)→0 as n→∞ The limit for the ratio test is 0.
=2(1200)/70=240/7=34 2/7 mph
The quadratic equation 5x2+7x-6=0 has discriminant 72-4(5)(-6)=49+120=169, it has both its roots real.
Since we are so near the solution, the quadratic formula gives x=(-7±13)/10=-2, 6/10
N.B. 5×4+7×(-2)-6=0 as does 5×(3/5)2+7×3/5-6=5×9/25+21/5-6=0
Let e = ounces of 11% alloy
t=ounces of 24% alloy
e +t=130 total weight of alloy
.11e+.24t=.16×130=20.8 total weight of copper
e=number of ounces of 11%Cu, t=number of ounces of 24% Cu
e+ t=130 total weight of alloy
.11e+.24t=.16×130=20.8 total weight of Cu
let u=x1/4. The equation now reads
u2-5u+6=0, a quadratic. Factor it and get
(u-3)(u-2)=0 or u=3, u=2
Solving for x we have x1/4=3→x=81
(1+secθ)/secθ = (1+1/cosθ)/(1/cosθ)
multiply numerator and denominator by cosθ to get
We are done.
Let u be the units digit t be the tens digit
The original number has value of u+10t
The number with digits reversed has value t+10u
The equations are now t + u = 12
For each x in the equation substitute (x-18).
For each y in the equation substitute (y+10).
e.g. y=x2 is a parabola through (0,0)
y+10=(x-18)2 is that same parabola translated to the right 18 units and down 10 units.