The normal to the curve y = x^3 - 5x2 +3x +1, at the points a(4, - 3) and b(1, 0) meet at point c. a) Find the coordinates of C. b) Find the area of triangle ABC?

9514 1404 393

  a) (-7, -2)

  b) 15 square units

Step-by-step explanation:

The slope at x is given by the derivative:

  y' = 3x^2 -10x +3

The slope of the normal is the negative reciprocal of this.

at a(4, -3) the slope of the normal is ...

  ma = -1/y' = -1/((3(4) -10)(4) +3) = -1/11

Then the point-slope equation of the line is ...

  y +3 = -1/11(x -4)

__

at b(1, 0) the slope of the normal is ...

  mb = -1/y' = -1/((3(1) -10)(1) +3) = -1/-4 = 1/4

Then the point-slope equation of the line is ...

  y = 1/4(x -1)

Solving these two equations will give the coordinates of point C.

  (1/4(x -1) +3) = -1/11(x -4)

  11(x -1) +132 = -4(x -4)

  15x = -105

  x = -7

  y = 1/4(-7 -1) = -2

The coordinates of point C are (-7, -2).

__

There are a few ways to find the area of a triangle that is specified by its vertex coordinates. I like the method that involves finding the determinants of pairs of coordinates around the figure. The area is half the absolute value of their sum. This can be made a little easier by listing the coordinates, repeating the first pair:

  a(4, -3)

  b(1, 0)

  c(-7, -2)

  a(4, -3)

Working down the list, the area will be ...

  A = 1/2|(4(0) -1(-3)) +(1(-2) -(-7)(0)) +((-7)(-3) -4(-2))| = 1/2|(0 +3) +(-2 +0) +(21 +8)|

  A = 1/2|30| = 15 . . . . square units

_____

The attached graph finds the intersection of the normals quite easily. The attached spreadsheet does the area calculation from the triangle coordinates. These tools are very handy for problems like this.

9) i can't see the angle so whatever it is just do this: cos(angle) x hypotenuse and then use that answer and they hypotenuse and plug it into this: (answer you just got)^2 + b^2 = (hypotenuse)^2. the first answer will be the answer of the adjacent side and the second answer after this equation will be the opposite side.

10) y = 14 x = √147 or 12.12435565

11) x = 4.8

12) x = 4.2

13) x = 9.3

14) pq: 13.2

qr : 17.6

angle p: 53º

15) angle c: 54.3º

angle a: 35.7º

16) h = 21040

v = 1841

step-by-step explanation:

10) 7/sin(30) = 14 = y

7^2 + b^2 = 14^2

49 + b^2 = 196

b^2 = 196 - 49

√b^2 = √147

x = √147

11) cos(70) x 14 = 4.788282007

x = 4.8

12) tan(25) x 9 = 4.196768923

x = 4.2

13) sin(31) x 18 = 9.270685348

x = 9.3

14) pq: sin(37) x 22 = 13.23993051

pq: 13.2

qr: cos(37) x 22 = 17.56998122

qr : 17.6

angle p: 180 - (37 + 90) = 53º

15) cos-1(7/12) = 54.31466529

angle c: 54.3º

angle a: 180 - (90 + 54.31466529) = 35.68533471

angle a: 35.7º

16) 4 x 5280 = 21120

cos(5) x 21120

h = 21039.63202

h = 21040

sin(5) x 21120

v = 1840.729287

v = 1841