Question

60. Modeling Data Property bounded by two perpendicular roads and a stream is shown in the figure. All distances are measured in feet. 600,i (0,540) (150, 430) (50, 390) (200,425) 400 (250, 360) (i0o, 390)(300. 275) 200 (350, 125) (400, 0) 200 400 600 (a) Use the regression capabilities of a graphing utility to fit a fourth-degree polynomial to the path of the stream. (b) Use the model in part (a) to find the area of the property in acres.

Answer 1

data

y	x	x^2	x^3	x^4
540	0	0	0	0
390	50	2500	125000	6250000
390	100	10000	1000000	1E+08
430	150	22500	3375000	5.06E+08
425	200	40000	8000000	1.6E+09
360	250	62500	15625000	3.91E+09
275	300	90000	27000000	8.1E+09
125	350	122500	42875000	1.5E+10
0	400	160000	64000000	2.56E+10

Excel regression result

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.99883899
R Square	0.997679327
Adjusted R Square	0.995358654
Standard Error	11.42630351
Observations	9
ANOVA
	df	SS	MS	F	Significance F
Regression	4	224516.6472	56129.16181	429.9095034	1.61316E-05
Residual	4	522.2416472	130.5604118
Total	8	225038.8889
	Coefficients	Standard Error	t Stat	P-value	Lower 95%
Intercept	536.9269619	11.17494794	48.04737926	1.12259E-06	505.9003324
x	-4.832252007	0.444379711	-10.87415084	0.00040595	-6.06604788
x^2	0.049632479	0.00496577	9.994920289	0.00056311	0.03584529
x^3	-0.000180378	1.92209E-05	-9.384490571	0.000718336	-0.000233744
x^4	1.95338E-07	2.38347E-08	8.195514646	0.001207603	1.29162E-07

y^ = 536.9270 -4.8323 x +0.0496 x^2 -0.000180378 *x^3 +1.95338*10^(-7) x^4

b)

$\int _0^{400}\:\:\:536.9270\:-4.8323\:x\:+0.0496\:x^2\:-0.000180378\:\cdot \:x^3\:+1.95338\cdot 10^{\left(-7\right)}\:x^4\:dx = 131953.15733\dots$