each letter in represents a different digit?

Question

1. Each letter in the folowing multiplication represents a different digit. Find the value of each letter.
A B C D E
_______4
E D C B A

2. I also want to ask how would you:
Sketch the region in the plane consisting of all points (x,y) such that
abs(X) + abs(y) is less than or equal to 1

3. Find the area of the region between the two concentric circles shown in the figure.
The figure shows two concentric circle. The ends of a line tangent to the inner circle touches the outer circle. This line has a length of 2.

Answer 1

1. Since your answer has five digits, A can only be 1 or 2. E*4 will be an even number, so A must be 2. E is 8 or 9. 9*3 would give a=6, which is wrong; so E=8. B*4 then cannot carry a 1, so B must be 0 or 1. If B=0 then 4D must end in 7, which it cannot; so B = 1. 4D ends in 8 and D cannot be 2; so D is 7. Trial and error shows that C has to be 9. In Summation, ABCDE = 21978.

2. Can't add a graph here, but your x values are only -1 to 1; and your y values similarly -1 to 1. Picture a diamond running from (0,1) to (1,0) to (0,-1) to (-1,0) and back to (0,1). The line would represent where abs(x)+abs(y)=1; so it would be a solid line. Everything within the diamond would be correct solutions, so that portion would be shaded.

3. A tangent line to the inner circle touches the outer circle at points such that the length of the line segment within the outer circle is 2. You have a right triangle then where one leg is r (the inner circle radius) and 1 (the distance from the tangent point to the outer circle); and the hypotenuse is x (the outer circle radius). x = sqrt(r^2+1).
The area of the inner circle is PIr^2.
The area of the outer circle is PIx^2
= PI(sqrt(r^2+1))^2
= PI(r^2+1)
= PIr^2 + PI

The difference between them is (PIr^2 + PI) - PIr^2 = PI
(PI being pi, or 3.141592653....)

Hope this all makes sense.

Answer 2

Since 1 was answered, here are 2 and 3.

2) its a square with corners at (1,0), (0,1), (-1,0), and (0, -1)

3) pi
call the radii a and b for the smaller and larger circles respectively. the area is pi(b^2 - a^2). Now if you draw a right triangle by connecting the midpoint and one of the endpoints of the chord to the center of the circles. Using the pythagorian theorem, you can see that b^2 - a^2 = 1. Thus the area of the donut is pi.

Answer 3

hi First C might desire to be < 4. If C became > 4, the tip result might have 4 digits (414 x 4 = 1656) C isn't a million, as CxC = E and C and E are distinctive C is two or 3 C x D is < 10. If no longer, there could be a document of a million, and the tip result does no longer be E. it would supply E + a million interior the extra severe case C = 2, and C x D < 10 ==> D < 5 ==> D < 6 desire it help Bye !

Answer 4

The first problem was easy enough but the others will have to be drawn to show the correct answers. Unfortunately my scanner isn't working otherwise I'd try to offer some more assistance.

A=2
B=1
C=9
D=7
E=8