Answer:
a) 25.15
b)
x = 1
y = t
z = (4pi)^2 + t *(8pi) = 4pi(4pi + 2t)
c) (x,y) = (1, -2pi)
Step-by-step explanation:
a)
First lets calculate the velocity, that is, the derivative of c(t) with respect to t:
v(t) = (-sin(t), cos(t), 2t)
The velocity at t0=4pi is:
v(4pi) = (0, 1, 8pi)
And the speed will be:
s(4pi) = √(0^2+1^2+ (8pi)^2) = 25.15
b)
The tangent line to c(t) at t0 = 4pi has the parametric form:
(x,y,z) = c(4pi) + t*v(4pi)
Since
c(4pi) = (1, 0, (4pi)^2)
The tangent curve has the following components:
x = 1
y = t
z = (4pi)^2 + t *(8pi) = 4pi(4pi + 2t)
c)
The intersection with the xy plane will occurr when z = 0
This happens at:
t1 = -2pi
Therefore, the intersection will occur at:
(x,y) = (1, -2pi)
If you find the discriminant it will tell you the number and types of roots. The discriminant is the value b^2 -4ac.
a = 1
b = 1
c = 1
1^2 - 4*1*1
1-4 = -3
Since this is a negative number there will be 2 complex roots.
Answer:
cost to make paper signs = $1
cost to make laminated signs = $2
Step-by-step explanation:
Let
x = cost to make paper signs
y = cost to make laminated signs
8x + 2y = 12 (1)
10x + 10y = 30 (2)
Multiply (1) by 5
40x + 10y = 60 (3)
10x + 10y = 30 (2)
Subtract (2) from (3) to solve for x
40x - 10x = 60 - 30
30x = 30
x = 30/30
x = $1
Substitute x = 1 into (1)
8x + 2y = 12
8(1) + 2y = 12
8 + 2y = 12
2y = 12 - 8
2y = 4
y = 4/2
y = $2
cost to make paper signs = $1
cost to make laminated signs = $2
The answer is 56.4 to the nearest tenth
