We write the equation in terms of dy/dx,
<span>y'(x)=sqrt (2y(x)+18)</span>
dy/dx = sqrt(2y + 18)
dy/dx = sqrt(2) ( sqrt(y + 9))
Separating the variables in the equation, we will have:
<span>1/sqrt(y + 9) dy= sqrt(2) dx </span>
Integrating both sides, we will obtain
<span>2sqrt(y+9) = x(sqrt(2)) + c </span>
<span>where c is a constant and can be determined by using the boundary condition given </span>
<span>y(5)=9 : x = 5, y = 9
</span><span>sqrt(9+9) = 5/sqrt(2) + C </span>
<span>C = sqrt(18) - 5/sqrt(2) = sqrt(2) / 2</span>
Substituting to the original equation,
sqrt(y+9) = x/sqrt(2) + sqrt(2) / 2
<span>sqrt(y+9) = (2x + 2) / 2sqrt(2)
</span>
Squaring both sides, we will obtain,
<span>y + 9 = ((2x+2)^2) / 8</span>
y = ((2x+2)^2) / 8 - 9
3364
= 2^2 * 29^2
So
√3364 = √(2^2 * 29^2) = 2 * 29 = 58
Steps to solve:
2(n + 3) = 2n + 3
~Distribute left side
2n + 6 = 2n + 3
~Subtract 6 to both sides
2n = 2n - 3
~Subtract 2n to both sides
n = -3
Best of Luck!
Let d = number of dimes, q = number of quarters
<span>Sally has 20 coins in her piggy bank, so </span>
<span>d + q = 20 </span>
<span>The total amount of money is $3.05. </span>
<span>Use the value of each coin multiplied by its number to get total value </span>
<span>.10d + .25q = 3.05 </span>
<span>multiply everything by 100 to clear the decimal places </span>
<span>10d + 25q = 305 and </span>
<span>d + q = 20 </span>
<span>solve the second equation above in terms of either d or q - lets do d </span>
<span>d = 20 - q </span>
<span>sub that into the first equation </span>
<span>10d + 25q = 305 </span>
<span>10(20 - q) + 25q = 305 </span>
<span>200 - 10q + 25q = 305 </span>
<span>15q = 105 </span>
<span>q = 105/15 = 7 </span>
<span>and from above </span>
<span>d = 20 - q = 20 - 7 = 13 </span>
<span>so, there are 13 dimes and 7 quarters </span>
<span>verify with orig problem </span>
<span>13 + 7 = 20, OK, and $1.30 + $1.75 = $3.05
</span>
An example for #1 would be that: 15 is divisible by 3, but not 9. every third multiple of 3, (9, 18, 27,...) is divisible by 9 because 9 is three times the size of 3.
For part two the first number that I thought of is 23.