A particle moves along the x −
axis in such a way that its acceleration at time
t
for
t
>
0
is given
a
(
t
)
=
3
t
2
. When
t
=
1
, the position of the particle is
6
and the velocity is
2.

(a) Write an equation for the velocity,
ν
(
t
)
, of the particle for all
t
>
0
.

(b) Write an equation for the position,
x
(
t
)
, of the particle for all
t
>
0
.

(c) Find the position of the particle when
t
=
e
. Give an exact answer.

It looks like the particle's acceleration is

a(t) = 3t²

and we're given the initial velocity and position to be v(1) = 2 and x(1) = 6.

(a) By the fundamental theorem of calculus, the velocity at time t is

v(t) = v(1) + ∫₁ᵗ a(u) du

v(t) = 2 + ∫₁ᵗ 3u² du

v(t) = 2 + u³ |₁ᵗ

v(t) = 2 + (t³ - 1³)

v(t) = t³ + 1

(b) Use the theorem again to get the position at time t,

x(t) = x(1) + ∫₁ᵗ v(u) du

x(t) = 6 + ∫₁ᵗ (u³ + 1) du

x(t) = 6 + (1/4 u⁴ + u) |₁ᵗ

x(t) = 6 + (1/4 t⁴ + t - 1/4 • 1⁴ - 1)

x(t) = 1/4 t⁴ + t + 19/4

(c) The position of the particle at t = e is

x(e) = 1/4 e⁴ + e + 19/4

a net is a diagram that shows a “flattened” view of a solid. in a net, each face and base is shown with all of its dimensions. a net can also serve as a pattern to build a three-dimensional solid.

a pyramid is a three-dimensional object with a base that is a polygon and triangular faces that meet at one vertex.

" the net of a triangular pyramid consists of four triangles. the base here we are given of the pyramid is a equilateral   triangle, and the lateral faces are also triangles "

spinning a 1, 2, or 3. option d.

step-by-step explanation: when spinning a number graph up to 4, and aiming for any number less than 3, that automatically eliminates the number 4. now you have two numbers to spin on that are less than 3, but the number 3 is included in this question because it's no greater of an number than itself. i hope this you : )