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here's a hint...try working backwards...

I'll try to help you as soon as possible...

Hey there! Calculating the monthly deposit required to accumulate $90,000 in 16 years with an APR of 5% is an important step towards building a college fund for your child.

To determine the monthly deposit, we can use the formula for the future value of an ordinary annuity:

FV = P * [(1 + r)^n - 1] / r

Where: FV is the future value ($90,000), P is the monthly deposit, r is the monthly interest rate (APR divided by 12), and n is the total number of months (16 years multiplied by 12 months).

Plugging in the values, we have:

$90,000 = P * [(1 + 0.05/12)^(16*12) - 1] / (0.05/12)

To solve for P, we can rearrange the equation:

P = ($90,000 * (0.05/12)) / [(1 + 0.05/12)^(16*12) - 1]

Performing the calculations, the monthly deposit required to accumulate $90,000 in 16 years at an APR of 5% is approximately $321.35.

\(a^3-b^3=(a-b)(a^2+ab+b^2)\\ sub \;\; a-b=4\qquad and \qquad a^3-b^3=52\\ 52=4(a^2+ab+b^2)\\ 13=(a^2+ab+b^2)\\ 13-3ab=a^2+ab+b^2-3ab\\ 13-3ab=a^2-2ab+b^2\\ 13-3ab=(a-b)^2\\ 13-3ab=4^2\\ -3ab=3\\ ab=-1\\~\\ 13=(a^2+ab+b^2)\\ 13+ab=(a^2+ab+b^2+ab)\\ 13+ab=(a^2+2ab+b^2)\\ 13+ab=(a+b)^2\\ 13-1=(a+b)^2\\ 12=(a+b)^2\\ a+b=\pm\sqrt{12}\)

So now check that i have not made any careless errors and solve simultaneously to get all possible values of a and b

Latex:

a^3-b^3=(a-b)(a^2+ab+b^2)\\
sub \;\; a-b=4\qquad and \qquad a^3-b^3=52\\
52=4(a^2+ab+b^2)\\
13=(a^2+ab+b^2)\\
13-3ab=a^2+ab+b^2-3ab\\
13-3ab=a^2-2ab+b^2\\
13-3ab=(a-b)^2\\
13-3ab=4^2\\
-3ab=3\\
ab=-1\\~\\
13=(a^2+ab+b^2)\\
13+ab=(a^2+ab+b^2+ab)\\
13+ab=(a^2+2ab+b^2)\\
13+ab=(a+b)^2\\
13-1=(a+b)^2\\
12=(a+b)^2\\
a+b=\pm\sqrt{12}

Like so:

This is a classic combinatorics problem that requires careful consideration of the given constraints. It can be quite challenging at first, but with some patience and a systematic approach, you can arrive at the correct solution. Good luck!

The equation for the height of the cannonball is:

h(t) = -4.9t^2 + 14t - 0.4

We want to find the time t when the height of the cannonball is 6 meters, so we set the equation equal to 6:

-4.9t^2 + 14t - 0.4 = 6

Solving for t, we get:

t = 1/2, 4

The first solution, t = 1/2, represents the time when the cannonball is first launched. The second solution, t = 4, represents the time when the cannonball reaches a height of 6 meters for the last time.

Therefore, the cannonball is above a height of 6 meters for a total of 4 - (1/2) = 7.5 seconds.

The total mass of the gold bars is 100%. So, 30% of the total mass is 0.3 * 100% = 30%. This means that the 17 lightest bars have a total mass of 30%.

Similarly, the 25 heaviest bars have a total mass of 48%. So, the remaining 22% of the total mass is the mass of the bars that Blair received.

Since the mass of each bar is the same, the number of bars that Blair received is 22% / 1% = 22 bars.

Therefore, Blair received 22 bars.

S==stamps

S - (0.2S + (t/7)S ==303, solve for S

S ==10605 / [28 - 5t]

Let t ==5

S==10605 / 3

S ==3,535 stamps in her 3 books

2/10 x 3535 ==707 stamps in her 1st book

5/7  x  3535 ==2525 stamps in her 2nd book.

And 303 stamps in her 3rd book.

Total ==707 + 2525 + 303 ==3,535 stamps in her collection.

2^n + 1 = 26

2^n =26 - 1

2^n =25

Take the log of both sides

n = log(25) / log(2)

n = ~ 4.643856......

D==Distance to beach,  T==Time it takes to drive to the beach

D/40 = T, 

D/25 = T + 3/4, solve for D, T

D==50 miles - distance to the beach.

T==5/4 ==1 hour + 15 minute - time it takes to drive to the beach at 40 mph. 

Oh, I see how it works now   

                                                      area₁  x  area₂  x  area₃     

                                                     (4 x 3)  x  (4 x 2)  x  (3 x 2)    

remove parentheses                      4 x 3   x   4 x 2   x   3 x 2      

rearrange and group                     (4 x 3 x 2)   x   (4 x 3 x 2)  

                                                                   24  x  24    

That's how it happens  

_.

5000 / 2000 x 35 = 87.5 minutes.

Put this very simple Python code in your Notebook and "run" it:

count = 0

for i in range(1, 1000):
    count += len(str(i))

print("Total digits:", count)

Adding  3a+mb to both sides of the second equation, we have
3a+2b=2
9a+(m+2)b=k
We multiply the first equation by 3, in order to make the coefficient of a equal to 9:
9a+6b=6
9a+(m+b)b=k
This system will have infinitely many solutions when both equations are the same. This occurs when k=6 and m+2=6.... 
im sure you can figure out what m is :)))

I don't know if it's a coincidence,   

but if you mutiply those areas                        12 x 8 x 6 = 576   

then take the sqrt of the product                    sqrt(576) = 24    

that's the same as the volume.  

_.

What is wrong with you? Can't you count?

From 1 to 9 =9 digits

From 10 to 99 =90 numbers  x 2 = 180 digits

From 100 to 999 = 900 numbers  x  3 =2,700 digits

Total =9 + 180 + 2700 =2889 digits.

P.S. Your coding skills in Python are pathetic!

Find the volume of a rectangular solid if the areas of three of its adjacent faces are 12, 8, 6  

If the dimensions are 4 x 3 x 2 you get those areas.   

I can't figure out another way to get the same areas.  

So the volume would be 4 x 3 x 2  =  24 cubic units   

_.

The rate of each team member is 2000 square feet / 35 minutes = 57.14 square feet per minute. The time it takes to paint a 5000 square foot wall is 5000 square feet / 57.14 square feet per minute = 87 minutes. The 25 members would take 87 minutes to paint the 5000 square foot wall. So the answer is 87

(a) Here is the Python code to calculate the number of digits:

def number_of_digits(n): if n < 10: return 1 elif n < 100: return 2 elif n < 1000: return 3 else: return 4 def total_number_of_digits(n): total_digits = 0 for i range(1, n + 1): total_digits +== number_of_digits(i) return total_digits print(total_number_of_digits(999))

The output of the code is 3515.  Therefore, the answer is 3515 digits.