Answer:
y-2 = 1/4(x-20) point slope form
y = 1/4x -3 slope intercept form
Step-by-step explanation:
y = -4x
The slope is -4
The slope that is perpendicular is the negative reciprocal
- (1/-4) = 1/4
The slope of our new line is 1/4
We have a point and the slope, so we can use point slope form
y-y1 = m(x-x1)
y-2 = 1/4(x-20)
If we want the line in slope intercept form, distribute
y-2 = 1/4x -5
Add 2 to each side
y-2+2 = 1/4x -5+2
y = 1/4x -3
C) -23x + 12 .. You're going to multiply the 4 to what's inside the parenthesis. So you'll get 12-24x+x Then just add the like terms which are the x's. 12-23x or -23x+12
The trick here is the recognize that the diagonal length of the rectangle is the diameter of the circle.
W = width of rectangle = 4
L = length of rectangle = 3
D = diagonal of rectangle
Using pythagorean theorem, we solve for D.
W^2 + L^2 = D^2
(4)^2 + (3)^2 = D^2
16 + 9 = D^2
25 = D^2
5 = D or -5 = D
Since D is a length, D must be positive. Therefore, D=5.
D = diagonal of the rectangle = 5
Since D is also the diameter of the circle AND the diameter of a circle is twice the radius, we have the following equation :
r = radius of circle = 1/2 D = 1/2 (5) = 2.5
C = circumference
C = 2 pi r = 2 pi (2.5) = 5 pi
Answer:
remember the chain rule:
h(x) = f(g(x))
h'(x) = f'(g(x))*g'(x)
or:
dh/dx = (df/dg)*(dg/dx)
we know that:
z = 4*e^x*ln(y)
where:
y = u*sin(v)
x = ln(u*cos(v))
We want to find:
dz/du
because y and x are functions of u, we can write this as:
dz/du = (dz/dx)*(dx/du) + (dz/dy)*(dy/du)
where:
(dz/dx) = 4*e^x*ln(y)
(dz/dy) = 4*e^x*(1/y)
(dx/du) = 1/(u*cos(v))*cos(v) = 1/u
(dy/du) = sin(v)
Replacing all of these we get:
dz/du = (4*e^x*ln(y))*( 1/u) + 4*e^x*(1/y)*sin(v)
= 4*e^x*( ln(y)/u + sin(v)/y)
replacing x and y we get:
dz/du = 4*e^(ln (u cos v))*( ln(u sin v)/u + sin(v)/(u*sin(v))
dz/du = 4*(u*cos(v))*(ln(u*sin(v))/u + 1/u)
Now let's do the same for dz/dv
dz/dv = (dz/dx)*(dx/dv) + (dz/dy)*(dy/dv)
where:
(dz/dx) = 4*e^x*ln(y)
(dz/dy) = 4*e^x*(1/y)
(dx/dv) = 1/(cos(v))*-sin(v) = -tan(v)
(dy/dv) = u*cos(v)
then:
dz/dv = 4*e^x*[ -ln(y)*tan(v) + u*cos(v)/y]
replacing the values of x and y we get:
dz/dv = 4*e^(ln(u*cos(v)))*[ -ln(u*sin(v))*tan(v) + u*cos(v)/(u*sin(v))]
dz/dv = 4*(u*cos(v))*[ -ln(u*sin(v))*tan(v) + 1/tan(v)]
