Consider the differential equation y prime(t)minus2y(t)equals10, y(0)equals4. a. Find a power series for the solution of the differential equation. b. Identify the function represented by the power series. a. Choose the correct answer below. A. y(t)equals4 plus 18 t plus 18 t squared plus 12 t cubed plus font size decreased by 4 times times times plus StartFraction 18 times 2 Superscript n minus 1 Over n exclamation mark EndFraction t Superscript n Baseline plus font size decreased by 4 times times times B. y(t)equals10 minus 18 t plus 18 t squared minus 12 t cubed plus font size decreased by 4 times times times plus StartFraction (negative 1 )Superscript n Baseline 18 times 2 Superscript n minus 1 Over n exclamation mark EndFraction t Superscript n Baseline plus font size decreased by 4 times times times C. y(t)equals4 minus 18 t plus 18 t squared minus 12 t cubed plus font size decreased by 4 times times times plus StartFraction (negative 1 )Superscript n Baseline 18 times 2 Superscript n minus 1 Over n exclamation mark EndFraction t Superscript n Baseline plus font size decreased by 4 times times times D. y(t)equals10 plus 18 t plus 18 t squared plus 12 t cubed plus font size decreased by 4 times times times plus StartFraction 18 times 2 Superscript n minus 1 Over n exclamation mark EndFraction t Superscript n Baseline plus font size decreased by 4 times times times b. y(t)equals nothing

a) $ 9.50

b) 17

c) y-intercept (0, 12). this is the initial amount of money that joe had prior to playing any game.

d) x-intercept ( 24, 0). this is the number of games that joe will play before he depletes the initial amount of money that he had.

e) slope is -0.5. joe's balance will reduce by 0.5 for every game that he plays.

f)

domain; [0,24]

range; [0,12]

g) x + 2y = 24

step-by-step explanation:

a)

if joe plays 5 games, this will imply that x = 5. substitute x = 5 in the equation and solve for y;

y = -.50x + 12

y = -0.50(5) + 12

y = 9.50

b)

if joe leaves the arcade with $3.50, this will mean that y = 3.50. plug in y = 3.50 in the given equation and solve for x;

y = -.50x + 12

3.50 = -0.50x + 12

0.50x = 12 - 3.50

0.5x = 8.5

x = 17

c)

the function given is   y = -.50x + 12.

the y-intercept is the point where the graph of this function will intersect the y-axis. at this point, the value of x is usually 0. plug in x = 0 in the given equation and determine y;

y = -0.50(0) + 12

y = 12

the y-intercept is thus (0, 12). this is the initial amount of money that joe had prior to playing any game.

d)

the x-intercept refers to the point where the graph of a function intersects with the x-axis. at this point, the value of y is 0. substitute y = 0 in the given equation and solve for x;

0 =   -.50x + 12

0.5x = 12

x = 24

x-intercept ( 24, 0). this is the number of games that joe will play before he depletes the initial amount of money that he had.

e)

the slope of the given equation is -0.5. the equation has been given in slope-intercept form. thus the slope is simply the coefficient of x.

joe's balance will reduce by 0.5 for every game that he plays. this is the interpretation of the slope.

f)

domain is the set of x values for which the function is real and defined. the domain will thus be; [0,24]. this will restrict the balance to only assume positive values.

on the other hand, the range is the set of y values for which a function is real and defined. the range in our case will be [0,12]. joe can have between $0 and $12 balance.

g)

the standard form of the equation of a straight line is;

ax + by = c

where a, b, and c are integers.

our equation is given as;

y = -.50x + 12

re-writing the equation we have;

0.5x + y = 12

divide all through by 0.5;

x + 2y = 24

step-by-step explanation:

when the line is solid:

when the line is dashed:

this > inequality sign means:

this < inequality sign means:

this > inequality sign means:

this < inequality sign means:

about the problem:

srry i'm not done ! i'll