Help with these math problems?

Question

I am stuck on these math problems. If you know how to do one, could you solve it but please explain the steps so I can understand how to do them! Thank you!!

Find an expression for a cubic function f if f(3) = 144 and f(-5) = f(0) = f(4) = 0.

Recent studies indicate that the average surface temperature of the earth has been rising steadily. Some scientists have modeled the temperature by the linear function T = 0.02t + 8.25, where T is temperature in °C and t represents years since 1900. Use the equation to predict the average global surface temperature in 2030.

Biologists have noticed that the chirping rate of crickets of a certain species is related to temperature, and the relationship appears to be very nearly linear. A cricket produces 113 chirps per minute at 70°F and 173 chirps per minute at 80°F.
(a) Find a linear equation that models the temperature T as a function of the number of chirps per minute N.
(b) What is the slope of the graph?
(c) If the crickets are chirping at 120 chirps per minute, estimate the temperature.

The table gives the winning heights for the Olympic pole vault competitions in the 20th century.
Year Height (ft)
1900 10.83
1904 11.48
1908 12.17
1912 12.96
1920 13.42
1924 12.96
1928 13.77
1932 14.15
1936 14.27
1948 14.1
1952 14.92

Year Height (ft)
1956 14.96
1960 15.42
1964 16.73
1968 17.71
1972 18.04
1976 18.04
1980 18.96
1984 18.85
1988 19.77
1992 19.02
1996 19.42
(a) Find and graph the regression line. (Let 1900 be x = 0. Enter any decimals correct to three places.)
(b) Use the linear model to predict the height of the winning pole vault at the 2000 Olympics.

(c) What is the difference between the predicted value and the actual winning height of 19.36 ft?

Thanks so much!

Answer 1

f(x) = -6x^3 -6x^2 + 120x



T(130) = 0.02(130) + 8.25 = 10.85



(a) T = (N + 155)/3



(b) 1/6



(c) 71.2°



(a) y = 0.089x + 11.087



(b) 19.999 ft



(c) 0.639 ft

Answer 2

1. cubic function is f(x)=ax^3+bx^2+cx+d

f(3) = 144--> f(x)= 144, x=3 thus 144=27a+ 9b+ 3c+ d

f(-5) = f(0) = f(4) = 0 ---> f(x)=0 for x= -5,0,4 thus you have: 0= -125a + 25b - 5c + d

and: 0= d

and: 0= 64a + 16b + 4c + d



With d=0 now you can set up a matrix of 3 equation and 3 variable a,b,c:



144=27a+ 9b+ 3c

0= -125a + 25b - 5c

0= 64a + 16b + 4c



These can also be writen as:



9a + 3b + c = 48

-25a + 5b - c = 0

16a + 4b + c =0



Solve for this matrix you will have a= -6, b=-6, c= 120

So the function is f(x)= -6x^3 - 6x^2+120x



2) T = 0.02t + 8.25, t is the represents years since 1900, find T in 2030.

t = 2030-1900=130



T= 0.02*130 + 8.25 = 10.85 °C



3) a) linear equation is T=mN + b (T as function of N)

113 chirps per minute at 70°F thus T= 70 and N= 113 ---> 70=113m +b

and 173 chirps per minute at 80°F. thus T= 80 and N= 173 --> 80= 173m+b



70=113m +b (1)

80= 173m+b (2)



subtract equations : (2) - (1) you will have 10=60m ,thus m= 1/6

plug m= 1/6 into equation (1) to find b : 70=113*1/6 +b , thus b= 307/6



therefore the function is: T= (1/6) N + (307/6)

b) the slope of the graph is m=1/6



c) N=120. (plug this into the equation to find T). Thus T= 120/6 + (307/6) = 71.2°F



4) this problem you only need to plug in points for the graph, x= height and y = years. This should be a piece of cake . Also use these values to set up a linear equations ( sould be similar to above problems)

b) then use the graph that you draw to find x with y= 2000, or you could use the equation

c) use the graph to find the predicted value of y for x= 19.36. and compare this number to the actual value.

Answer 3

I need help with this math problem.



"The height of the winning pole vault in the early years of the modern Olympic Games can be modeled as a function of tine by the formula.



H = 0.08t + 3.2



Here t is the number of years since 1900, and H is the winning height in meters. (One meter is 39.37 inches.)



(a) Calculate H(12).( Round your answer to two decimal places.)

H(12)=____________m"