Step-by-step explanation:
h = height of rectangle (should be "length", but ok)
w = width of rectangle
w = h + 97
the diagonal of the rectangle is the Hypotenuse (baseline) of a right-angled triangle with the legs being one height and one width side of the rectangle.
so, we use Pythagoras
c² = a² + b²
with c being the Hypotenuse (the side opposite of the 90° angle).
diagonal² = h² + (h + 97)² = h² + h² + 2×97h + 97²
113² = 12769 = 2h² + 194h + 9409
3360 = 2h² + 194h
1680 = h² + 97h = (h + 97/2)² - 9409/4
(completing the square)
6720/4 + 9409/4 = (h + 97/2)²
16129/4 = (h + 97/2)²
127/2 = h + 97/2
30/2 = h = 15
the height is 15.0 in
Answer:
The increase is 1150.
11.5 times the original value, or 1150% of it. That's a big increase, and I would not buy that thing, whatever it is.
Answer:
x=3, y= -2
Step-by-step explanation:
Multiplying :
(4x+3y =6)*-3
(3x+5y =-1)*4
-12x -9y= -18
12x+20y= -4
Add the equations
20y-9y = -4-18
11y = -22
y= -22/11 = -2
3x +5* -2= -1
3x -10 = -1
3x = 10-1
x= 9/3 =3
He's using the graduated cylinder method. I think the volume might be between 15 and 20. (It's kind of not obvious...)
Short AnswerThere are two numbers
x1 = -0.25 + 0.9682i <<<<
answer 1x2 = - 0.25 - 0.9582i <<<<
answer 2 I take it there are two such numbers.
Let one number = x
Let one number = y
x + y = -0.5
y = - 0.5 - x (1)
xy = 1 (2)
Put equation 1 into equation 2
xy = 1
x(-0.5 - x) = 1
-0.5x - x^2 = 1 Subtract 1 from both sides.
-0.5x - x^2 - 1 = 0 Order these by powers
-x^2 - 0.5x -1 = 0 Multiply though by - 1
x^2 + 0.5x + 1 = 0 Use the quadratic formula to solve this.
a = 1
b = 0.5
c = 1
x = [-0.5 +/- sqrt(0.25 - 4)] / 2
x = [-0.5 +/- sqrt(-3.75)] / 2
x = [-0.25 +/- 0.9682i
x1 = -0.25 + 0.9682 i
x2 = -0.25 - 0.9682 i
These two are conjugates. They will add as x1 + x2 = -0.25 - 0.25 = - 0.50.
The complex parts cancel out. Getting them to multiply to 1 will be a little more difficult. I'll do that under the check.
Check(-0.25 - 0.9682i)(-0.25 + 0.9682i)
Use FOIL
F:-0.25 * -0.25 = 0.0625
O: -0.25*0.9682i
I: +0.25*0.9682i
L: -0.9682i*0.9682i = - 0.9375 i^2 = 0.9375
NoticeThe two middle terms (labled "O" and "I" ) cancel out. They are of opposite signs.
The final result is 0.9375 and 0.0625 add up to 1