#4**+2 **

2b^2 + b = 6

step 1.... divide both sides by 2

b^2 + (1/2)b = 3

step 2 .....take (1/2) of (1/2) = 1/4....square this = 1/16 add to both sides

b^2 + 1/2b + 1/16 = 3 + 1/16

step 3 ......factor the left side.....simplify the right side

(b + 1/4)^2 = 49/16

step 4 ......take positive/negative roots of both sides

b + 1/4 = ±√[ 49/16]

b + 1/4 = ±7/4

step 5......subtract 1/4 from both sides

b = 7/4 - 1/4 = 6/4 = 3/2 or b = -7/4 - 1/4 = -8/4 = -2

The mistake was in step 2.....1/4 was added to both sides, but it should have been (1/4)^2 = 1/16 !!

CPhill Aug 23, 2017

#5**+2 **

Here's 22

f(x) = -0.1x^2 + 3.2x -3.5

Let's find the roots....thus

-0.1x^2 + 3.2x -3.5 = 0 multiply both sides by -10

x^2 - 32x + 35 = 0 subtract 35 from both sides

x^2 - 32x = -35 take 1/2 of 32 = 16....square this = 256 ....add to both sides

x^2 - 32x + 256 = -35 + 256 factor and simplify

(x - 16)^2 = 221 take positive/negative roots of both sides

x - 16 = ±√221 add 16 to both sides

x = ±√221 + 16

So....the two roots are -√ 221 + 16 ≈ 1.13 and √ 221 + 16 ≈ 30.87

And the positive difference between these two is the width of the tunnel at the bottom =

30.87 - 1.13 ≈ 29.74 ft....so....the tunnel is wide enough at the bottom

The function is an upside-down parabola....the vertex occurs at the max height

To find the height...let's find the x coordinate of the vertex.....this is given by

-b / (2a) where b = 3.2 and a = -.1 ....so we have

-3.2 / [ 2(-.1) ] = -3.2 / -.2 = 32 / 2 = 16

So....putting this into the original function will give us the height....and we have

-.1(16)^2 + 3.2(16) - 3.5 =

-25.6 + 51.2 - 3.5 = 22.1 ft

However.......we need to find the width at 20 feet

So....set the function = 20 and solve

20 = -0.1x^2 + 3.2x -3.5 subtract 20 from both sides

Using the quadratic formula, we have that

x = ( -3.2 ±√ [ 3.2^2 - 4(-.1)(-23.5) ] ) / [ 2(-.1)] =

x = (-3.2 ±√ [ .84] ) / [ -.2 ] =

x ≈ 11.41 and x ≈ 20.58

So...the width at 20 feet equals the distance between these ≈ 9.17 feet

So....the tunnel is not wide enough at this point.....!!!!

CPhill Aug 23, 2017