Bài tập CALCULUS 51

Created by T. Madas
Question 181

(****)
y

(

y = 1 x 2 − 12 x + 35
4

)

R
L1
Q

P

O

x

S
L2
The figure above shows the curve with equation

(

)

y = 1 x 2 − 12 x + 35 .
4
The curve crosses the x axis at the points P ( x1, 0 ) and Q ( x2 ,0 ) , where x2 > x1 .
The tangent to the curve at Q is the straight line L1 .

a) Find an equation of L1 .
The tangent to the curve at the point R is denoted by L2 . It is further given that L2
meets L1 at right angles, at the point S .

b) Find an equation of L2 .
c) Determine the exact coordinates of S .

(

C1Q , y = 1 x − 7 , 4 y + 8 x = 31 , S 9 , − 5
2
2
2 4

Created by T. Madas

)


Created by T. Madas
Question 182

(****)

y

P

y = 11 − x 2 −

Q

9

x2

S

R

O

y =1

x

The figure above shows the curve C with equation

y = 11 − x 2 −

9

x2

, x ≠ 0.

The straight line with equation y = 1 meets C at the points P , Q , R and S , where
R and S have positive x coordinates, as shown in the figure.
Find the area of the finite region bounded by C and the line segment RS .
16
3

Created by T. Madas


Created by T. Madas
Question 183

(****)

y

x

y

x
The figure above shows the design for an earring consisting of a quarter circle with two
identical rectangles attached to either straight edge of the quarter circle. The quarter
circle has radius x cm and the each of the rectangles measure x cm by y cm .
The earring is assumed to have negligible thickness and treated as a two dimensional
object with area 12.25 cm 2 .

a) Show that the perimeter, P cm , of the earring is given by

P = 2x +

49
.
2x

b) Find the value of x that makes the perimeter of the earring minimum, fully
justifying that this value of x produces a minimum perimeter.
c) Show that for the value of x found in part (b), the corresponding value of y is
7 4 −π ) .
16 (

MP1-H , x = 3.5

Created by T. Madas