The sum of the elements of any row [ after the first one] is always 1 greater than the sum of the elements of all the previous rows
To see this....note that the sum of elements on any row [ starting with n = 0] is just 2n
Note that, for example, the sum of the first 4 rows is
20 + 21 + 22 + 23 = 1 + 2 + 4 + 8 = 15 = 24 - 1
And adding 1 to this sum = 16 = 24 which is just the sum of the 5th row elements
y = 32 bounded above and below since this is a horizontal line
y = 2x bounded below by the x axis.....unbounded above
y = 2 - x2 this is an inverted parabola with a vertex at (2,0)....thus...it is bounded above and unbounded below
y = √ [ 1 - x2 ] this is the upper part of a circle with a radius of 1.....thus....it is bounded below and above
y = x2 √ [ x + 4 ] ......this one is hard to see......look at the graph here :
Notice that it is bounded below but not above
ab = 7
a^2b + ab^2 + a + b = 80
What is a^2 + b^2 ???
So we have :
a^2b + ab^2 + a + b = 80 factor the first two terms
ab (a + b) + (a + b) = 80 factor again
( a + b) (ab + 1) = 80
(a + b) (7 + 1) = 80
(a + b) (8) = 80 divide by 8 on both sides
a + b = 10
Square both sides
a^2 + 2ab + b^2 = 100
a^2 + 2(7) + b^2 = 100
a^2 + 14 + b^2 = 100 subtract 14 from both sides
a^2 + b^2 = 86
For f(x), write y
y = ax^2 + bx + c subtract c from both sides
y - c = a [ x^2 + (b/a)x ]
Complete the square on x....take (1/2) of (b/a) = b/[2a]....square it = b^2/[4a^2 ]........add to both sides.....don't forget that we're actually adding ab^2/[4a^2 ] to the left side
y - c + ab^2 / [4a^2] = a [ x^2 + (b/a)x + b^2/(4a^2 ) ]
Factor the right side........simplify the left
y - c + b^2 / (4a) = a [ x + b /(2a) ] ^2
Add c, subtract b^2 / (4a) to both sides
y = a [ x + b/(2a) ] ^2 - b^2 / (4a) + c
y = a [ x + b / (2a) ]^2 + ( - b^2 / (4a) + c )
So.....the vertex is given by [ -b / (2a) , - b^2 / (4a) + c ]
Note that we can prove that this is correct.........
The x coordinate of the vertex is -b / (2a)
Putting this into ax^2 + bx + c for x, we can find the y coordinate of the vertex thusly :
y = a [b^2 / (4a^2) ] + b [ -b/(2a)] + c
y = [b^2 / (4a)] - b^2 /(2a) + c
y = [ b^2 - 2b^2 ] / (4a) + c
y = [-b^2 / (4a) ] + c
Last one :
A photographer offers a photo shoot for a $85 flat fee. Customers may purchase prints for $5 per sheet.
How many sheets can a customer purchase and spend at most $150?
What linear inequality with variable x represents this situation?
What is the solution to that inequality? Enter the solution as an inequality using x.
For the first one....we have this inequality
85 + 5x ≤ 150 where x is the number of sheets
Subtract 85 from both sides
5x ≤ 65
Divide both sides by 5
x ≤ 13 .....so the customer can by 13 sheets at most
Two cars leave the same location traveling in opposite directions. One car leaves at 3:00 p.m. traveling at an average rate of 55 miles per hour. The other car leaves at 4:00 p.m. traveling at an average rate of 75 miles per hour. Let x represent the number of hours after the first car leaves.
How many hours after the first car leaves will the two cars be 380 miles apart?
We want to find this : distance traveled by first car + distance traveled bt second car = 380
And distance = rate * time
So...the first car travels x hours and the second car travels one hour less = x - 1
And we have
55x + 75(x - 1) = 380 simplify
55x + 75x - 75 = 380 add 75 to both sides
130x = 455 divide both sides by 130
x = 3.5 hrs
Jason started a hat-making business. He spent $750 to purchase supplies to get started and he uses about $3.50 worth of supplies per hat made. Jason charges $15 for each hat. Let h represent the number of hats.
What is the minimum number of hats Jason will need to sell to make a profit?
He will make a profit when Sales > Costs ...so
15h > 750 + 3.50h subtract 3.50h from both sides
11.5h > 750 divide both sides by 11.5
h > ≈ 66 hats
m = [ a + b] / 2 multiply both sides by 2
2m = a + b subtract b from both sides
2m - b = a
That's all I have time for....!!!!