please help me with a math home-work problem and show me how you derived at your answer?

Question

Joe's annual income has been increasing each year by the same amount. The first year his income was 24,000, and the ninth year his income was 31, 680. In which year was his income 45,120.

Answer 1

31,680 - 24,000 = 7,600
7,600/8 = 960

45,120 - 31,680 = 13,440
13,440/960 = 14

9 + 14 = 23

In year 23, his income would be 45,120

Answer 2

I think that the easiest way to do this purely by geometry is to use the theorem of Pappus-Guldinus. This says that any volume that can be generated by sweeping a plane shape along a path is equal to the product of the area of the plane shape and the length of the path of the centroid of the plane shape. I will show you how it applies here then explain how the theorem can be derived from geometry alone. In the case of a cone, the plane shape is a right triangle of height h and radius r. The centroid, or balance point of a right triangle is located 1/3 the way from the right angle to the other vertices. If you put the right angle at an origin, then the centroid is at (r/3, h/3). To generate a cone, revolve this triangle about the vertical side of length h. This means that the path of the centroid is a complete circle of radius r/3 giving a path length of 2π(r/3). The area of the triangle is rh/2. The volume is the product of these two: V = 2π(r/3) * rh/2 = π r^2 h/3 You can find the volume of many complex shapes using this theorem. The theorem works because of the definition of the centroid. If you imagine the plane shape following some curving path, it is easy to show that, as it plane bends back and forth, the volume lost by the part inside the curve exactly matches the volume added outside the curve. This statement is only true if the path followed is always on the centroid of the shape. You can find the centroid of a triangle by the geometric construction of drawing lines from each vertex to the center of the opposite side. Both the theorem of Pappus-Guldinus and the determination of the centroid were done by geometry long before calculus was invented so the volume of a cone by geometry alone is well documented.

Answer 3

first you have to make an equation(i screwed up the equation so the answer is probably wrong but ill tell you how to find it anyway in case you know the equation)

2400 + 31680x=45120(all i know is that the 2nd part of the equation is right but the first half isnt)
then you have to get the variable on one side by itself and what to do to one side of the equation, you have to do to the other side.

2400-2400(and put a box around it to show that it leavs) + 31680x=45120-2400

so now the equation is 31680x=42720

then you have to get the x by itself so you have to find the reciprical of 31680. so you have to find the common denominator of 31680 which is always 1 because 1 goes into everything but if its a fraction then you just flip it

so 31680/1 so the reciprical is 1/31680.
42720/1(you need a common denomenator for that too but not the reciprical) X 1/31680= 42720/31680 which is 1.348485

Answer 4

Let x be the amount by which his income increased each year. Since his income in the ninth year is 31680, and his income for the 1st year was 24000, x must satisfy the equation:

24000 + 8x = 31680, since his income increased 8 times from the 1st yr to the ninth yr. Solving for x gives x=960.

Let n be the number of years (after the first year) that were required for his income to reach 45120. Then n must satisfy the equation:

24000 + 960n = 45120, or n =22.

His income was therefore 45120 in the (n+1)th year, or the 23rd year.

Answer 5

since his income increases uniformly each year,
the equation would be:

present income = 24,000 + (increase rate * [yrs of service-1])
use [yrs of service-1] because let's say at 9yrs he will have his 8th salary increas..

get his increase rate,
*at 9yrs, and letting x = increase rate

31,680 = 24,000 + 8X
7,680 = 8x
x = 960...

therefore,

45,120 =24,000 + (960 * [yrs of service -1])
21,120 = 960 * (yrs of service -1)
22 = yrs of service - 1
yrs of service = 23

ANSWER is 23th year...
hope it helps :)