All Answers 355923 Answers

The fact that  BC is parallel to DE  sets up the following relationship :

AC/CE  = AB /BD

AC / 15  = 8 /10     

AC / 15 =  4 /5    multiply both sides by 15

AC  = 15 * 4  /5   =   60 / 5   = 12

The bisection of the apex angle sets up the following relationship :

DB / DA    =  BC / AC

x / 2.25   =   4 / 3      multiply both sides by 2.25

x  = 2.25 *  4  /3 

x  =  9/ 3

x  = 3 

sin  ( x + pi)

_________

cos ( x - pi)

sinx * cos pi  + sin pi * cos x

_______________________ 

cosx *cos pi + sinx * sin pi

 sin x * (-1) + (0) * cos x

___________________

oos x * (-1)  = sin x * (0)

-sin x

_____

-cos x

tan x

44.

We  have this function

y  = A cos (Bθ )

A  = the amplitude = l 0.5  l   = 0.5

To find B....we have that

2pi  / period  = B

2pi / [ 2 pi / 3 ]  = B

2pi * 3  / 2pi  = B 

3  = B

So...the correct answer is

y  = 0.5cos (3θ )

We have that

x /  4   =  40 / 32

x / 4   =  5 / 4       multiply both sides by  4

x = 4 ( 5/4)   =   5  cm

Notice that the perimeter of the second figure  / perimeter of the first figure will be the linear scale factor  =

34 / 20  =   17/10   = 1.7

So....the area of the second figure will be =

area of first figure  * (linear scale factor)^2  =

19.6  * (1.7)^2  =

56.6  m^2

Thank you!

We will have this system of equations

a(2)^2  + b(2)  + c  = 28.4

a(3)^2  + b(3) + c  = 63.9

a(4)^2  + b(4)  + c  = 113.6       simplifying, we have

4a + 2b + c  = 28.4

9a +  3b + c = 63.9

16a  + 4b + c = 113.6

Subtract the first equation from both the second and third equations and we have this :

5a +  b  = 35.5

12a + 2b  = 85.2

Multiply the  first equation by  -2  and add it to the second equation

-10a -2b  = - 71

12a + 2b  = 85.2

______________

  2a  =  14.2            divide both sides by  2

a  =  7.1

And using 5a + b = 35.5,  we can solve for b as

5 (7.1) + b = 35.5

35.5 + b  =35.5

b = 0

And using  4a + 2b + c  =28.8  to find c

4(7.1) + 2(0) + c  =28.8

28.8  + c  =28.8

c  = 0

So...our function  is

y  = (7.1 )x^2           and  b  and c  = 0

2: A steel sphere with a 3-inch radius is made by removing metal from the corners of a cube that has the shortest possible side lengths. How many cubic inches are in the volume of the cube?

The sphere will be tangent to each face of the cube.....so....the side length of the cube must  be  6 in

So....the volume of the cube  is   just

side^3   =   6^3   =  216 in^3

We use the formula:

\(A=\frac12ab\sin{C}\)

Therefore, the area of the triangle is:

\(\frac12\cdot14.2\cdot8\cdot\sin{99}\)

I hope this helped,

Gavin

1: If the diameter of a right cylindrical can with circular bases is increased by 25%, by what percent should the height be increased in order to double the volume of the original can?

For convenience sake...let the original diameter of the can  =2

So....the original radius  = 1

So...the original volume is

V = pi * (1)^2 * original height

V = pi * original height

original height = V /pi

When the original diameter is increased by 25%, so is the original radius...so the new radius is 1.25

And we have that

2V  = pi * (1.25)^2 * new height

[2V ] / pi * (1.25)^2  =  new height

So

new height

____________   =

original height

[2V] / [ pi *(1.25)^2 ]

_________________      =

   V /pi

      2V                           pi

__________    *     _________

pi * (1.25)^2                V

2 / (1.25)^2  =   

1.28

So....the original height should be increased by [ 1.28  - 1]  =  .28  = 28%    to double the volume

2)Find the distance between the graphs of 5x - 12y + 45 = 0 and 5x - 12y - 46 = 0.

The first line has a y intercept  of 

-12 y = -45

y = 45 / 12

y = 15/4  = 3.75

So....the point  (0, 3.75)  is on the first line

And the distance betwen this point and the second line will be the perpendicular distance between the two lines....so...we have

l 5 (0)  - 12 (3.75) - 46  l             l 91 l                    91

____________________  =    ___________  =   ___   =   7   units

   √[ 5^2 + 12^2]                         √169                    13

Thank you!

1)Find the distance between (3,4) and the line 4x + 3y + 7 = 0

We can use this to find the answer.....the distance between  (a, b)  and the line  AX + BY + C  = 0  is given by

l A(a)  +  B(b)  +  C  l   

_________________

  √ [ A^2 + B^2 ]

So we have

  l  4(3) + 3(4)  l         l  24  l               24

  ___________  =     _______  =      ___  =    4.8 units

  √ [ 4^2 + 3^2]           √25                  5

2sin x   cos x  + √3 cos x  = 0        factor out  cos x

cos x  (2 sinx + √3 )   = 0

Set each factor to 0 and solve for x

cos x  = 0    and this happens at   x = pi/2  and 3pi/2

2sinx + √3  =  0     subtract  √3 from both sides

2 sin x  = -√3       divide both sides by 2

sin x  = -√3 / 2     and this happens at  4pi/3 and  5pi/3

So...the correct answer is

pi/2, 4pi/3, 3pi/2, 5pi/3

Not sure about 'simplest form"   but

1:2   is   3:6

and

3:4    is    6:8

then you have

3:6:8

1st figure= 20m, 2nd figure=34m

20:34

1:34/20

Area=19.6

So, 19.6*34/20=\({33.32}=\boxed{33.3}\)

I believe it is "x√7−49√x"

By the Law of Sines,

\(\frac{\sin B}{8.2}=\frac{\sin 32°}{6.7}\\~\\ \sin B=\frac{8.2\sin 32°}{6.7}\\~\\ \begin{array}{lcl} B=\arcsin(\frac{8.2\sin 32°}{6.7})&\qquad\text{or}\qquad&B=180°-\arcsin(\frac{8.2\sin 32°}{6.7})\\~\\ B\approx40.433°&\text{or}&B\approx139.567° \end{array}\)

First let's use the first possible value of  B .  If  B = 40.433° ,  then...

C  =  180° - 32° - 40.433°

C  =  107.567°

And by the Law of Sines,

\(\frac{AB}{\sin107.567°}=\frac{6.7}{\sin32°}\\~\\ AB=\frac{6.7\sin107.67°}{\sin32°}\\~\\ AB\approx12.054\)

Second let's use the second possible value of  B .  If  B = 139.567° ,  then...

C  =  180° - 32° - 139.567°

C  =  8.433°

And by the Law of Sines,

\(\frac{AB}{\sin8.433°}=\frac{6.7}{\sin32°}\\~\\ AB=\frac{6.7\sin8.433°}{\sin32°}\\~\\ AB\approx1.854\)

So the two possible values of  AB  are   AB  ≈  12.054   and   AB  ≈  1.854

If   AB  =  12.054   then

the area of  △ABC  =  (1/2)(8.2)(12.054 sin 32° )

the area of  △ABC  ≈  26.189

If  AB  =  1.854   then

the area of  △ABC  =  (1/2)(8.2)(1.854 sin 32° )

the area of  △ABC  ≈  4.028

So....     AB  ≈  1.854

Hey NSN!

Find \(\csc{\frac{5\pi}{4}}\)

\(\csc{\frac{5\pi}{4}}\) = \(\frac{1}{\sin{\frac{5\pi}{4}}}\)

\(\sin{\frac{5\pi}{4}}=\sin({\frac{\pi}{4}+\pi})=-\frac{\sin{\pi}}{4}=-\frac{\sqrt{2}}{2}\)

\(\boxed{-\sqrt{2}}\)

I hope this helped,

Gavin

Thanks!

Welcome, CPhill! I like your solution!

44. ?

42. Assuming s is the major arc, here is the solution:

π here is 180 degrees, so 5π/3 is also 5 * 180 / 3 = 300

That angle is 300 degrees. 

The circumference is 2r * π, where r is the radius and π is 3.1415...

The circumference is 18π. 

Since the major arc is 300/360 or 5/6 of the circumference, it is 

5/6 * 18π = 15π long. 

This in decimal is 15 • 3.141...=47.124...

A. 

I hope this helped,

Gavin

45. tan 5pi/6=\(\frac{-\sqrt3}{3}\)