CPhill

We need to solve this

-16t^2 + 60t = 56     rearrange as

16t^2 - 60t + 56 = 0       divide through by 4

4t^2 - 15t + 14  = 0   factor

(4t - 7) (t - 2)  = 0

Set each factor to 0 and solve for t  and we get   7/4 sec  and 2 sec

7/4 sec is correct  = 1.75 sec

Look at the following image:

I agree with the guest....the inner circle is not needed

In the circle,  the angles E and D are bisected by OE and OD....so the measure of angles OED and ODE = 37.5°....so angle DOE = 105°

So....each chord will span an arc of 105°

Note that.... 

3 chords span 3(105) = 315°  = - 45°    and we are at point "D"

6 chords span 6(105)  = 630° = - 90°   and we are at point "G"

So....every 3 chords will end up shifting us  a multiple of -45° from "A" 

So...we need      360 / 45 * (3)  =  8 * 3  = 24  chords to bring us back to "A"

We can use the Law of Cosines to find the missing side, S

S^2  = 143^2 + 220^2 - 2(143)(220)cos(75.7)

S^2  = 53307.8

S = √53307.8  ≈  231 ft

a. Are the events "Chocolate" and "Adults" independent? Why or why not?

If independent.... P (Chocolate l Adult )  = P(Chocolate and Adult) /P(Chocolate)

                                      .21/ (.21 + .39 + .60)  =  .21/ .35  ???

                                       .21   =  .60   ????    not independent

b. Are the events "Children" and "Chocolate" independent? Why or why not?

P(Children l Chocolate)  =  P(Children and Chocolate) / P(Children)  ???

                        .14/ (.14+ .21+ .35)     =   .14 / .40  ???

                         .20  = .35  ???   not independent

c. Are the events "Vanilla" and "Children" independent? Why or why not?

P(Vanilla l Children)  = P(Vanilla and Children) / P(Vanilla)      ???    

      .26 / (   .14 + .26 + .40)  =  .26 / .65    ???

                  .325 = .40 ???    not independent

2) Alex from Ayden and Bob from Beaufort are driving towards each other at the same time. They meet at a first point 32km away from Ayden. After Alex arrives at Beaufort and Bob arrives ay Ayden, they turn around right away. They meet at a second point 64km away from Ayden. Find the distance from Ayden to Beaufort.

Let the distance be D

Let Alex's Rate = A

Let Bob's Rate = B

Distance / Rate = Time

When they meet the first time  Alex has traveld 32 km  and Bob has traveled (D - 32) km

So.....from the start until they meet the first time we have that the times are equal

32/ A  =  (D - 32) / B    ⇒   A/B  = 32 / (D - 32)

And after the first meeting.....Alex has traveled (D - 32 + D -64)  = (2D - 96)

And Bob has traveled (32 + 64) = 96

So......equating the time again

(2D -96) /A  = (96) /B ⇒   A/B  = (2D - 96) / 96

And A/B  =A/B   so

32 / (D -32)  = (2D -96) / 96     cross-multiply

96 * 32  = (2D - 96) ( D - 32)     simplify

96*32  = 2D^2 - 160D + 96*32

2D^2 - 160D = 0

2D ( D - 80)  = 0     so

2D = 0                                 D - 80  = 0  

D = 0   reject                       D = 80km

Here you go for the first one : https://www.desmos.com/calculator/ofobm3u4af

For the second

You have the correct graphs...the intersection points are  ( -1, -3)  and ( 3,5)

Algebraically....set the functions equal

x^2 -4  = 2x - 1       rearrange as

x^2 - 2x - 3  = 0   factor

(x - 3) ( x + 1)  = 0

Set each factor and solve for x

x -3  = 0          x + 1  = 0

x = 3                  x = -1

Put these values into either function for x and you should get the same intersection points as above

Hope that helps  !!!

a) Show that the locus of points specified by 𝑥 = 3 cos 𝜃 + 4 and 𝑦 = 3 sin 𝜃 − 2 is a circle.

cos θ =  (x - 4) /3        sin θ = (y + 2) / 3  

And

cos^θ  + sin^2θ = 1       ....so.....

(x - 4)^2 / 3^2  + (y + 2) /3^2  = 1

(x - 4)^2  + (y + 2)^2  = 3^2

(x -4)^2  + (y + 2)^2  = 9

This is a circle  cenrered at ( 4, -2)  with a radius of  3

y = -2x + 8   would work

In fact.....any equation in the form   y = -2x + b      would work

xy = 56   ⇒  y = 56/x

And

7 (1/x) +  14(1/y) = 4

7/x + 14/y  = 4

7/x + 14 / (56/x)  = 4

7/x + (14/56)x  = 4

7/x + (1/4) x  = 4          multiply through by  4x

28 + x^2 = 16x           rearrange as

x^2 - 16x + 28  = 0      factor as

(x - 14) ( x - 2)  = 0

Set each factor to 0  and solve for x   and we have that

x = 14     or x = 2 

And y =

56/14 = 4     or   56/2 = 28

So  (x,y)   = (2, 28)

Just calculate  38 x 8