Answer:
f(x) = 1.5x - 0.5x
Step-by-step explanation:
The function of the pattern represented by the pentagonal numbers is the sum of three triangular numbers.
The triangular number general formula
x (x + 1) / 2
For example,
The sequence
1, 3, 6, 10
*
* * *
* * * * * *
* , * *, * * *, * * * *
_____________________________
The pentagonal numbers
The sequence:
1, 5, 12, 22, 35
As shown in the picture can be divided into three triangles
Triangle 2
x (x + 1) / 2
Triangle 1 and 3 (they are triangles one unit smaller than 2)
n (n + 1) / 2
n= x-1
Replacing n
(x-1) ((x-1) + 1) / 2
(x-1) (x) / 2
(x-1) x / 2
______________
Function represents the pattern
Triangle 2 + (Triangle 1 + Triangle 3)
Triangle 1 = Triangle 3
So then,
Triangle 2 + 2* Triangle 1
x (x +1) /2 + 2* (x -1) x/2
Rearranging
0.5 x (x +1) + x(x -1)
0.5x^2 + 0.5x + x^2 -x
(0.5 x^2 + x^2) + (0.5x -x )
1.5 x^2 - 0.5 x
______
Y = 4x + 3 ---------(1)
y = -x - 2 ---------(2)
Subtract (2) from (1), you get
0 = 5x + 5
⇒ 5x + 5 = 0
⇒ 5x = -5
⇒ x = -5/5 = -1
y = -(-1) - 2 = -1
Answer is (-1,-1)
OK, so the graph is a parabola, with points x=0,y=0; x=6,y=-9; and x=12,y=0
Because the roots of the equation are 0 and 12, we know the formula is therefore of the form
y = ax(x - 12), for some a
So put in x = 6
-9 = 6a(-6)
9 = 36a
a = 1/4
So the parabola has a curve y = x(x-12) / 4, which can also be written y = 0.25x² - 3x
The gradient of this is dy/dx = 0.5x - 3
The key property of a parabolic dish is that it focuses radio waves travelling parallel to the y axis to a single point. So we should arrive at the same focal point no matter what point we chose to look at. So we can pick any point we like - e.g. the point x = 4, y = -8
Gradient of the parabolic mirror at x = 4 is -1
So the gradient of the normal to the mirror at x = 4 is therefore 1.
Radio waves initially travelling vertically downwards are reflected about the normal - which has a gradient of 1, so they're reflected so that they are travelling horizontally. So they arrive parallel to the y axis, and leave parallel to the x axis.
So the focal point is at y = -8, i.e. 1 metre above the back of the dish.
The answer is 13 hope this helps
The volume formula is V= l x L x H, l=width, L=Length, H= Depth, so
2x3 _ 9x2 + 7x + 6 = l x L x (2x + 1), because H=(2x + 1), so
l x L= (2x3 _ 9x2 + 7x + 6 )/ (2x + 1) = (2x3 _ 9x2 + 7x + 6 ) X [1/(2x + 1)]
case1: l= (2x3 _ 9x2 + 7x + 6 ) or L= 1/(2x + 1), case2: L= (2x3 _ 9x2 + 7x + 6 ) or l= 1/(2x + 1)
the why question:
perhaps there is similarity of value between volume and l, or volume and L
