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Let  x  be the number of minutes it would normally take to drive the route

Since

R * T  = D     and  D  is the same in both cases, we have that

30 ( x + 18)  = 45 (x  - 8)        simplify

30 x + 540  = 45 x - 360     rearrange as

540 + 360 = 45x - 30x

900 = 15x  divide both sides by 15

60  = x   

So....the distance  is  30 (60 + 18)  = 30 mph ( 78  min)   

But  we need to  convert 78 min to hours =  78/60  = 39/30 hr

So...the total distance is  30m/h ( 39/30 h)  = 39 miles 

HAHAHA!!!...I'm only a "genius" at sporadic periods.....!!!!!

Here's a pic :

Wow, that is smart. Maybe you are the genius and not me, HAHA

Since two of the sides must be parallel, assume that AB  is parallel to CD

Drop a perpendicular from C to AB and let the intersection be the point E

Let EB  = x

Similarly, drop a perpendicular from D to AB  and let the intersection be the point F

Let FA  = 52 - (39 + x)  = 13 - x

So....the height  of the triangles  = height of the trapezoid

The  height of the triangle on the right  =√[12^2 - x^2] = CE

The height of the triangle on the left is √ [5^2 - (13 - x)^2 ] = DF

And, as stated,  CE  = DF

Equating the quantities under the roots, we have

12^2 - x^2  = 5^2  - (13 - x)^2

144 - x^2 = 25 - (169 - 26x + x^2)

144 - x^2 = 25 - 169 + 26x - x^2

144  = 25 - 169 +26x

144 = -144 + 26x

288 = 26x

x  = 288/26  = 144/13

So...the  height of the trapezoid = √[12^2 - (144/13)^2 ] = 60/13

And the area  =  (1/2)(height)(sum of the bases)  =

(1/2)(60/13) ( 39 + 52)  =

(1/2) (60/13) (91)  =

(1/2)(60) (91/13)  =

(1/2)(60)(7)  =

(1/2)(420)  =

210 units^2 

The question:

"ABCD is a trapezoid with the measure of base AB twice the measure of the base CD. Point E is the point of intersection of the diagonals. The measure of diagonal AC is 11. Find the length of segment EC. Express your answer as a common fraction."

∠AEB  and  ∠DEC  are vertical angles, so they are congruent.

∠CAB  and  ∠ACD  are alternate angles, so they are congruent.

By the AA similarity theorem, △ABE ~ △CDE, so...

AE / AB   =   CE / CD

The measure of diagonal AC is 11.

AE + CE  =  AC

AE + CE  =  11

AE  =  11 - CE

And the measure of base AB is twice the measure of the base CD.

AB  =  2CD

Now we can use these values of  AE  and  AB .

AE / AB  =  CE / CD

                                                     Substitute  2CD  in for  AB  and  11 - CE  in for  AE.

(11 - CE) / (2CD)  =  CE / CD

                                                     Multiply both sides of the equation by  CD .

(11 - CE) / 2  =  CE

                                                     Multiply both sides of the equation by  2 .

11 - CE   =   2CE

                                                     Add  CE  to both sides.

11  =  3CE

                                                     Divide both sides by  3 .

11 / 3  =  CE

EC  =  11 / 3

Where is the picture?

3x  - 2y  = 6    rearrange as

3x - 6  = 2y     divide through by 2

(3/2)x - 3  = y

A line perpendicular to this will have a slope of (-2/3)

If  this line has a y intercept  of 2, then the point (0,2)  is on the graph

So......the equation of the line is

y  = (-2/3)x + 2

To find the x intercept, let y  = 0

0  = (-2/3)x + 2

-2 = (-2/3)x

x  = -2(-3/2)  = 3

Here's a graph : https://www.desmos.com/calculator/fi2c8ubbdw

Look at this problem in a much simpler and intuitive way as follows:

When "b" = 5 goes to "b"= 3, that is a direct decrease of 3/5 =0.6, when c=9

And since "b" is directly proportional to "a" and inversely proportional to "c", it follows that:

New "c" = 9 x 1/0.6 =15

a.   If the side of the square is 7 times longer, then the area  is 7^2  = 49 times more

So.....49 * 1000   = 49000

a.  We have an octagon

There are 8 congruent triangles each  with an included angle of 45° between sides of  13.07 in

So...the total area  = 8 (1/2)(13.07)^2 (sin 45°)  ≈ 483 in^2

a. We have a trapezoid with  one base of 6    and a height of  8

The other base  = 6 + 2 + 8  (the triangle on the left has the same base as height )  =  16

The area  is  (1/2)height * (  sum of bases) =

(1/2)(8)(22)   =

88 cm^2

a. For the triangle on the right....the base is opposite a 30° angle so  its length is 1/2 * 6  = 3ft

The height of the triangle [ and the trapezoid]  =  3√3 ft

So...the area of the trapezoid  =

(1/2)(height) (sum of the bases)  =

(1/2)(3√3)(11 + 14)  =

(1/2)(3√3)(25)  =

(3/2)(25)√3 =

(75/2)√3 ft^2

Hint: ABE and CDE are similar when drawing the diagram.

1. For 4ft by 4ft, 1000 earthworms can fit, so for a square has a side length 7 times longer, the answer is 49,000 feet, or (a).

2. This shape is an 8-sided figure or an octagon. This is the longest diagonal, so the answer is 483.2 inches, rounded to the nearest tenth: 483.2 inches, or (A).

5(last one): Plotting, we have: https://www.desmos.com/calculator/bm72xzzt3z, we can split this figure into a rectangle and a triangle: 3*4=12-rectangle, and 1/2*4*4=8, so 12+8=20 units.

Thank you!

- Daisy

2 = 5k
k=2/5 constant of direct proportionality. But:............................(1)
2 =k/9
k=18 constant of inverse proportionality. But when:..................(2)
b =3, from (1) above:
a =2/5 x 3=6/5 = 1.2.  Using (2) above for the new value of a:
1.2 =18 / c
c = 18 / 1.2
c =15 when b = 3 

Square  with a "radius"  of 11m

The "radius"  is the distance from the center of the square to any vertex

The area is  4(1/2)11^2  = 2 * 121  = 242 m^2

Pentagon with a side of 9m

The pentagon will be composed of 5 congruent triangles

The height of each is given  by  (9/2)/tan (36°)

So...the total area is

5 (1/2)(9)(9/2)/tan(36°)  =  (5)(9/2)^2/tan(36°)  ≈  139.4 m^2

Pentagon  with a radius of 5m

We have 5 congruent triangles with sides of 5 and an included angle of 72° between these sides

The area  =   5 (1/2) (5)^2sin (72°) ≈  59.4m^2

Hexagon with a side of 12 yd

We have six congruent equilateral triangles...each side  = 12 yd

The area is  6(1/2)(12)^2* √3/2  ≈  216√3  yds^2 ≈ 374.1 yds^2

Since this parabola turns upward....1 will still be in the range when, at a minimum,  (m , 1)  is the

vertex

We have the form   ax^2 + bx + c

To  find "m"  we   can use    -b/ [2a]  =       5 / (2 * 1)   =  5/2  = m

So  (5/2, 1)  is the vertex

So....subbing this into the function for x  and f(x)  to find " c,"  we have

1  = (5/2)^2  - 5(5/2) + c

1 = 25/4 - 25/2 + c

1  = 25/4 - 50/4 + c

1  = -25/4  + c

1 + 25/4  = c

29/4   =  c

Here's a graph to show this : https://www.desmos.com/calculator/gbhwg6nerl

 Check for yourself that if c > 29/4,  then  1 will be out of range

Despite Melody saying she didn't have time to write up and check her answer, what she wrote was nevertheless almost complete. The only omission was in the "Let M be the midpoint of AC" bit. For the sake of those who want a complete proof this bit should say:–

"Since AD is parallel to BC we must have CAD = ACB (as indicated in the diagram, but without justification). Drop a perpendicular BM from B to AC. At this stage we dont know that M is the midpoint of AC. But the two triangles AMB, CMB are congruent (angle, angle, side). Therefore AM=MB."

The second "proof" isn't really a proof at all, because it assumes that \(\alpha=45^\circ\), which isn't necessary at all, as Melody has shown.

By the way, we've never heard of a trapezoid in Mathematics in the UK (including England) either. It is the universal name of a bone in your hand, so if you were up in Anatomy you would have heard of it :-) The only name in the UK for a 4-sided figure with no parallel sides is a quadrilateral (admittedly an irregular one).

Thank you!!

Thanks, Daisy.....!!!...here's another way to see this

Break-even point is where  Costs  = Revenue

Let the number of screwdrivers that need to be sold  = S

So...Total  Costs = 124  +  1S

Total Revenue  = 3S

Set these even

124 + 1S  = 3S     subtract 1S from both sides

124  = 2S    divide both sides by 2

62  =  S   ⇒   62 screwdrivers must be sold to break even

Guest, your last 4 answers are correct!

Your question is missing some important numbers! Check it carefully.

Thank you so much for your help!! Are the last questions correct? I worked them out but I'm not 100% sure if they are right. Sorry.

Our profit per package delivered is \($2.70 - $1.70 = $1\). Since we need to make \($15\), we can do \(\frac{15}{1} = 15\). So we would need \(15\) packages to be delivered.

- Daisy

The profit of the screwdriver is \($2.00\), and we need to sell enough screwdrivers to make a profit of \($124.00\). We can solve this by doing \(\frac{124}{2} = 62\). So you need to have \(62\) screwdrivers produced.

- Daisy