Suppose r(t)=costi+sintj+3tk represents the position of a particle on a helix, where z is the height of the particle above the ground.

a. What is t when the particle has height 16?
b. What is the velocity of the particle when its height is 16?
c. When the particle has height 16, it leaves the helix and moves along the tangent line at the constant velocity found in part b. Find a vector parametric equation for the position of the particle (in terms of the original parameter t) as it moves along this tangent line.

a. The component tells you the particle's height:

b. The particle's velocity is obtained by differentiating its position function:

so that its velocity at time is

c. The tangent to at is

the system of inequity is

step-by-step explanation:

from the given graph it is noticed that we have two related lines. one is solid and another is dashed.

if a line passing through two points, then the equation of line is

the solid line passing through the points (0,4) and (4,0).

the related equation of solid line is .

similarly, find the related equation of dashed line. the dashed liune passing through the points (0,0) and (3,1).

the related equation of dashed line is

from the given graph it is noticed that the point (0,1) lies on the shaded region, therefore the point (0,1) must be satisfied by the related equations.

check the related equations by the point (0,1).

this statement is true if the sign of inequality is < instead of equality. since it is an solid line therefore the sign of inequality will be . the required inequality is

similarly, check the second related equation.

it is a dashed line, so this statement is true if the sign of inequality is > . the required inequality is