Hello!

The velocity is the derivative of the displacement function. Therefore we can find the displacement as the function of time by integrating the velocity function.

For v(t) =` t^3 - 11t^2 + 34t - 24` ,

the displacement d(t) = `(1/4)t^4 - (11/3)t^3 + 17t^2 - 24t + C` ,

where C is an arbitrary constant.

The displacement between t=1 and t=7 is equal to d(7) - d(1), which is equal to

[`(1/4)*7^4 - (11/3)*7^3 + 17*7^2 - 24*7 + C` ] -

[`(1/4)*1^4 - (11/3)*1^3 + 17*1^2 - 24*1 + C` ] =

(1/4)*(2401 - 1) - (11/3)*(343 - 1) + 17*(49 - 1) - 24*(7 - 1).

Further, this is equal to

2400/4 - 11*342/3 + 17*48 - 24*6 =

600 - 11*114 + 816 - 144 = **18 (feet)**.

This is the answer for **(a)**.

**(b)** is more complicated. Integrating v(t) sums up direct and back movements with respect to their directions, but to find the distance travelled we have to count direct and back movements separately.

Formally it is integral of |v(t)| (the absolute value of v(t)).

Less formally, when the particle moves only forward, its distance coincide with the displacement. Same when it moves only back.

So we have to determine intervals where the sign of v(t) is unchanged. The only points where sign of v(t) could change are its roots.

We can try possible integer roots (they are the dividers of the free term, 24) and discover that 1, 4 and 6 are the roots. Also we can build graph on the computer, see please https://www.desmos.com/calculator/xijfxw64jl

Now we integrate v(t) from 1 to 4, from 4 to 6 and from 6 to 7, and sum the absolute value of the results.

Distance = |d(4) - d(1)| + |d(6) - d(4)| + |d(7) - d(6)|. I omit C because it is cancels out as it should.

**d(1)** = (1/4)1^4 - (11/3)1^3 + 17*1^2 - 24*1 = 1/4 - 11/3 - 7 = **1/4 - 2/3 - 10**,**d(4)** = (1/4)4^4 - (11/3)4^3 + 17*4^2 - 24*4 = 4^3 - 704/3 + 176 = 240 - 704/3 = 240 - 234 - 2/3 = **6 - 2/3**,**d(6)** = (1/4)6^4 - (11/3)6^3 + 17*6^2 - 24*6 = 324 - 792 + 468 = **0**,**d(7)** = (1/4)7^4 - (11/3)7^3 + 17*7^2 - 24*7 = 2401/4 - 3773/3 + 665 = 600 + 1/4 - 1257 - 2/3 +665 = **8 + 1/4 - 2/3**.

The result is

|6 - 2/3 - (1/4 - 2/3 - 10)| + |0 - (6 - 2/3)| + |8 + 1/4 - 2/3 - 0| =

16 - 1/4 + 6 - 2/3 + 8 + 1/4 - 2/3 =

30 - 4/3 = 28 and 2/3 =** 28.66(6) (feet)**.

(I tried to double-check all computations, but it is always a possibility for a mistake)