Question

A small aircraft starts its descent from an altitude of mile, 4 miles west of the runway (see figure). 1 (a) Find the cubic f(x) - ax + bx? ox + d on the interval [-4, O that describes a smooth glide path for the landing. (b) The function in part (a) models the olde path of the plane when would the plane be descending at the greatest rate?

Answer 1

(a)

(-4.4/3) (0,0)

given f(x)=ax³+bx²+cx+d
=>f '(x)=3ax²+2bx+c
=>f ''(x)=6ax+2b

from the figure f(-4)=4_/3 , f(0)=0, f '(-4)=0 , f "(-2)=0

f "(-2)=0
=>6a(-2)+2b=0
=>-12a+2b=0
=>b=6a

f '(-4)=0
=>3a(-4)²+2b(-4)+c=0
=>48a-8b+c=0

f(0)=0
=>a(0)³+b(0)²+c(0)+d=0
=>d=0

f(-4)=4_/3
=>a(-4)³+b(-4)²+c(-4)+d=4_/3
=>-64a+16b-4c+d=4_/3

48a-8b+c=0,b=6a
=>48a-8(6a)+c=0
=>48a-48a+c=0
=>c=0

-64a+16b-4c+d=4_/3,b=6a,d=0,c=0
=>-64a+16(6a)-4(0)+(0)=4_/3
=>-64a+96a=4_/3
=>32a=4_/3
=>a=4_/3*32
=>a=1_/24

b=6a,a=1_/24
=>b=6*1_/24
=>b=1_/4

+ + - + 0x+0 1. => f(x) = 1+4

(b)

point of inflection occurs when x=-2

so plane will be descending at the greatest rate when x= - 2

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