1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
TiliK225 [7]
3 years ago
10

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?
Mathematics
2 answers:
Gnom [1K]3 years ago
6 0

9514 1404 393

Answer:

  a) (-7, -2)

  b) 15 square units

Step-by-step explanation:

<h3>(a)</h3>

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).

__

<h3>(b)</h3>

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.

Llana [10]3 years ago
6 0

Answer:

Step-by-step explanation:

Curve:\ y=x^3-5x^2+3x+1\\\\y'(x)=3x^2-10x+3\\A=(4,-3)\\B=(1,0)\\y'(4)=3*4^2-10*4+3=11\\slope\ of\ the\ normal\ in \ A: -\frac{1}{11} \\Equation\ normal\ in \ A:\\y+3= -\frac{1}{11}*(x-4)\ or\ -x-11*y=29\ (1)\\\\y'(1)=3*1^2-10*1+3=-4\\slope\ of\ the\ normal\ in \ B: \frac{1}{4} \\Equation\ normal\ in \ B:\\y-0=\frac{1}{4} (x-1)\ or\ -x+4y=-1\ (2)\\\\Coordinates\ of\ C:\\(1)-(2) ==> y=-2\\(2) ==> x=-7\\\\C=(-7,-2)\\

Area of the  triangle ABC:

S=\dfrac{|CB|*|CA|*sin(\widehat{ACB})}{2}\\A=(4,-3),B=(1,0),C=(-7,-2)\\AB^2=(4-1)^2+(-3-0)^2=9+9=18\\AC^2=122\\BC^2=68\\\\Using\ Al'Kashi\ theorem:\\AB^2=CB^2+CA^2-2|CB||CA|*cos(\widehat{ACB})\\\\cos(\widehat{ACB})=\dfrac{68+122-18}{2\sqrt{68*122} } =\dfrac{86}{\sqrt{68*122} } \\\\sin^2(\widehat{ACB})=1-cos^2(\widehat{ACB})=1-\dfrac{86^2}{68*122} \\\\S=\dfrac{\sqrt{68} *\sqrt{122}*\sqrt{\dfrac{900}{68*122}  }}{2}=15\\

You might be interested in
When baking a cake you have a choice of the following pans: a round cake that is 2 inches deep and has a 7-inch diameter, a 6 in
irga5000 [103]
The answer is
round cake - 82.42 in²
rectangular cake - 114 in²

Round cake:
d = 7 in
r = d/2 = 7 in / 2 = 3.5 in
h = 2 in

The surface are of a cylinder is:
A = 2πr² + 2πrh

The surface are of the round cake (which is actually a cylindrical cake) excluding the bottom is:
A = 2πr² + 2πrh - πr²
A = πr² + 2πrh
A = 3.14 * 3.5² + 2 * 3.14 * 3.5 * 2
    = 38.46 + 43.96
    = 82.42 in²

Rectangular cake:
w = 6 in
l = 9 in
h = 2 in

The surface are of a rectangle is:
A = 2wl + 2wh + 2lh

The surface are of the rectangular cake excluding the bottom is:
A = 2wl + 2wh + 2lh - wl
A = wl + 2wh + 2lh
A = 6 * 9 + 2 * 6 * 2 + 2 * 9 * 2
    = 54 + 24 + 36
    = 114 in²

5 0
3 years ago
Evaluate The following expression . in e^e
erik [133]

\bf \textit{Logarithm Cancellation Rules} \\\\ \stackrel{\stackrel{\textit{let's use this one}}{\downarrow }}{log_a a^x = x}\qquad \qquad a^{log_a x}=x \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ln(e^e)\implies log_e(e^e)\implies e

3 0
3 years ago
PLEASE HELP MEEEEE!!! IT's TIMED
Triss [41]

Answer:

x is less than oe equal to 12

5 0
2 years ago
Read 2 more answers
3y + 7 = x is this a direct proportion
Tju [1.3M]

Answer:

no

Step-by-step explanation:

if the ratio (yx) of two variables (x and y) is equal to a constant (k=yx) .in this case y is said to be directly proportional to x with proportionality constant k

3 0
2 years ago
From the trail head, the waterfall is 4/5 mile, and then the eagle’s nest is 3/4 mile farther. From there, it is 1 3/8 miles bac
Anvisha [2.4K]

Answer:

Need more info

Step-by-step explanation:

5 0
3 years ago
Other questions:
  • Solve for m: m/5 + 3(m-1)/) = 2(m-3)
    5·2 answers
  • What is the similarity ratio of ΔABC to ΔDEF?<br> Tri ABC: AC=4 CB=5<br> Tri DEF: DF= 2
    14·1 answer
  • 3) Is the following a function?<br> (-4, 10) (10,-6) (8, 10) (2,-9) (0,7)
    5·2 answers
  • Complete the following addition exercises.
    15·2 answers
  • A customer deposits $1275 in a savings account that pays 4% interest compounded quarterly. How much money will the customer have
    13·1 answer
  • A robot can complete 8 tasks in 5/6 hour. Each task takes the same amount of time.
    11·1 answer
  • Solve this Equation <br><br> -8 = 6 + ___
    10·2 answers
  • A circle is inscribed in a square. The circumference of the circle is increasing at a constant rate of 6 in/sec. As the circle e
    5·1 answer
  • The sum of two numbers is –40. One number
    6·1 answer
  • Determine a series of transformations that would map polygon ABCDE onto polygon A’B’C’D’E’?
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!