#### To determine

**Part (a):**

**To find:**

The cubic polynomial that best models the velocity data of the shuttle for the timeinterval tϵ[0,125] by using a graphing calculator or computer

#### Answer

vt=0.00146t3-0.11553t2+24.98169t-21.26872

#### Explanation

**1) Concept:**

Use a T1 84 calculator to find the best fit cubic model for the given data

**2) Given:**

The table gives the velocity data for the shuttle:

**3) Calculation:**

Use Cubic Reg option of graphing calculator (**TI 84 calculator)** to find the best-fitting model for Cubic polynomial

From the given data, for given time interval tϵ[0,125], vt∈0, 4151

**Steps to find cubic polynomial using TI 84 calculator:**

Step 1: Enter Stat Editor, then Press STAT.

Step 2: Select Edit to input data into lists.

Step 3: When there is data in list from previous work, it needs to be cleared before work is continued

Step 4: Enter data in L1 and L2.

Step 5: Press 2^{nd} STAT PLOT (this is the Y= key). At this point, need to check if the plots are on or off. All plots should be off except the one with which you are working.

Step 6: Press ENTER to select Plot 1.

Step 7: Turn Plot1 ON by highlighting the word ON and pressing ENTER.

Step 8: Select the ‘Cubic plot’ for plot Type.

Step 9: Select L1 for X list and L2 for Y list.

Substitute the values for time t in L1 and vt in L2.

By using T1 84 calculator the output is,

vt=0.00146t3-0.11553t2+24.98169t-21.26872

**Graph of**vt**:**

**Conclusion:**

The cubic polynomial that best models the velocity data of the shuttle for the timeinterval tϵ[0,125] by using aT1 84 calculatorcalculator is 0.00146t3-0.11553t2+24.98169t-21.26872

#### To determine

**Part (b):**

**To estimate:**

The height reached by the Endeavour 125 seconds after the liftoff using the model in part (a)

#### Answer

206,407 ft

#### Explanation

**1) Concept:**

i) the property of the Integral:

∫fx±gx dx=∫fx dx±∫gx dx

ii) The fundamental theorem of calculus part 2

If f is continuous on a,b then,

∫abfxdx=Fb-F(a)

Where F is any antiderivative of f, that is, the function F such that F'=f

**3) Calculation:**

The velocity data of the shuttle is the rate of change of height reached by the Endeavourfor the timeinterval

vt=h't

Here, t∈0,125

By the Net Change theorem,

∫0125vt dt≈∫0125h'tdt=h125-h(0) height reached by the Endeavour for the timeinterval

h125-h0=∫0125vt dt

=∫01250.00146t3-0.11553t2+24.98169t-21.26872 dt

By using the properties of integral,

=∫01250.00146t3 dt-∫01250.11553t2 dt+∫012524.98169t dt-∫012521.26872 dt

=0.00146t44-0.11553t33+24.98169t22-21.26872t0125

=0.000365t4-0.03851t3+12.490845t2-21.26872t0125

By applying the fundamental theorem of calculus,

=0.0003651254-0.038511253+12.4908451252-21.26872125-0

≈206,407 ft

**Conclusion:**

The height reached by the Endeavour 125 seconds after the liftoff using model in part (a) is 206,407 ft