All Answers 358501 Answers

1 - Will consider 5% APR as "compounded monthly".

2 - Use this formula to find the monthly payments.

3 -  FV=90,000; R=0.05/12; N=16*12;PMT = FV*(((1 + R)^N - 1)^-1* R)

4 -  PMT =$ 306.91 - monthly payments that must be made for 16 years or: 16 x 12==192 months.

i uh had a brain fart 

(1, 716 , 1145 , 4852 , 5391 , 5567 , 5996 , 6106 , 6535)^9 mod 7007==1 - there are 9 values of n.

 (1, 673 , 1249 , 1537 , 4225 , 4417 , 5185 , 6529 , 6625)^9 mod 7008==1 - there are 9 values of n

 (1, 216 , 221 , 466 , 737 , 853 , 1111 , 1468 , 1670 , 1683 , 2014 , 2530 , 3261 , 3476 , 3527 , 4215 , 4602 , 4860 , 4994 , 5419 , 5682 , 5763 , 6069 , 6279 , 6327 , 6787 , 6886)^9 mod 7009==1 - there are 27 values of n.

The total: 9 x 9 x 27 ==2,187 values of n

Check: taking any 3 random numbers such as: 5391, 4417, 3261 and applying Chinese Remainder Theorem, we have:

344,177,337,504 m +  339,715,607,617, where m=0, 1, 2, 3....etc. When m=0, then we have: 339,715,607,617, which is < than the LCM. And:

339,715,607,617^9 mod 7007==1
339,715,607,617^9 mod 7008==1
339,715,607,617^9 mod 7009==1

there are 38 woman and 17 like to dance so    so the prob is   17/38

All the other information is irrelevant.

I used a different method and arrived at the same result as you.

The square of the distance from the origin to the parabola is

\(\displaystyle x^{2}+\left\{\frac{1}{\sqrt{2}}(x^{2}-18)\right\}^{2}\)

\(\displaystyle =\frac{1}{2}(2x^{2}+x^{4}-36x^{2}+324) \\ \displaystyle =\frac{1}{2}(x^{4}-34x^{2}+324) \\ \displaystyle = \frac{1}{2}(x^{4}-34x^{2}+289+324-289) \\ \displaystyle = \frac{1}{2}\{(x^{2}-17)^{2}+35\}.\)

That has a minimum value of 35/2 occurring when x squared is equal to 17.

So the minimum distance is\(\displaystyle \sqrt{35/2}\)  .                                                                   

<<<< it appeared that the positive and negative root 17 were the minimum x values >>>>   

I approximated 1/sqrt(2) as 0.707 and used Desmos to graph y = 0.707(x² – 18)  

My parabola crossed the x-axis near +4.2 ... that's close enough to sqrt(17) that    

I'd say we got the same answer there.  

I believe, however, that that is not the shortest distance to the origin.  

If you look at the graph, you see that the curve is slanting outward and 

crosses the x-axis at an angle.  The measurement to that intersection  

from the origin would be a line on top of the x-axis itself.  

The shortest distance from the origin to the curve has to be the line that's     

perpendicular to the curve.  That's going to be somewhere below the x-axis.    

Eyeballing the graph on Desmos, which is an approximation, the point seems    

to be somewhere close to (4, –2) on the positive side.  I don't know how to do   

the calculations that would determine it accurately.   

_.

If the tax rate is set at 5%, calculate the total tax amount that a company would need to pay if they import goods worth $50,000 into the United States.   

                                      Tax  =  5% times $50,000    

                                      Tax  =  0.05 • $50,000     

                                      Tax  =  $2,500   

_.

A 25 foot ladder leans against a vertical wall. The foot of the ladder is 7 feet from the base of the wall. If the top of the ladder slips 4 feet down the wall, how far will the foot of the ladder slide?      

One way to do this is to use Pythagoras' Theorem twice.  Sort of a before and after.    

This would be much better if I could just draw the pictures.  But I can't, so here goes...   

Visualize the ladder as the hypotenuse of a right triangle.  

Then the hypotenuse is 25 and the ground side is 7.   

By the theorem a² + b² = c², then    a² (the wall)  =  c² (the ladder) – b² (the ground)        

                                                         a²  =  25² – 7²    

                                                         a²  =  625 – 49  =  576       

                                                         a  =  sqrt(576)  =  24 (the height of the wall side)      

When the ladder slips 4 feet down the wall, the new "a" becomes 20   

The hypotenuse stays the same 25 because it's a ladder   

So now we can say                         b² (the ground)  =  25² (the ladder) – 20² (the wall)         

                                                       b²  =  625 – 400  =  225    

                                                       b  =  sqrt(225)  =  15 (the length of the ground side)    

When the top of the ladder slid down the side of the wall 4 feet,

the bottom of the ladder slid out until it was 15 feet from the wall.   

The ladder was already 7 feet from the wall, so the bottom of the ladder slid 8 feet.   

_.

Can you rationalize the demoninator? 

Let k be the x co-ordinate of the point C, then its y co-ordinate (C above the x axis) will be sqrt(1 - k^2/4).

(The path of H will be symmetric about the x axis, so we need only bother with the part above the x axis. We can double the calculated area to get the total area.)

The altitude through C will be the line x = k.

The altitude through B will be perpendicular to the line AC.

The slope of AC is (sqrt(1- k^2 /4) - 0)/(k + 4), so the slope of BH will be -(k + 4)/(sqrt(1-k^2/4) - 0).

The equation of BH will be

\(\displaystyle \frac{y-0}{x-4}=-\frac{k+4}{\sqrt{1-k^{2}/4}}\)

This intersects the altitude through C at H, so the y co-ordinate of H will be  \(\displaystyle \frac{(4-k)(4+k)}{\sqrt{1-k^{2}/4}}=\frac{2(16-k^{2})}{\sqrt{4-k^{2}}}.\)

k varies from x = -2 to x = +2, so the area above the x axis determined by the path of H will be

\(\displaystyle \int^{2}_{-2}\frac{32-2x^{2}}{\sqrt{4-x^{2}}}dx.\)

Substitute \(x = 2\sin\theta,\)

and the integral becomes

\(\displaystyle \int^{\pi/2}_{-\pi/2}28+4\cos2\theta\:d\theta=28\pi.\)

That means that the total area will be (hopefully, check for mistakes,) \(56\pi.\)

Yes, there are two of them.

\((1+i)/\sqrt{2}\quad \text{ and } \quad-(1+i)/\sqrt{2}.\)

a was 5 and b was 4

We can solve this problem by converting the numbers to base 10 and then adding them.

The number (4AB)_6 in base 10 is 46^2 + A6^1 + B*6^0 = 144 + 6A + B.

The number (41)_6 in base 10 is 46^1 + 16^0 = 24 + 1.

The number (53A)_6 in base 10 is 56^2 + 36^1 + A*6^0 = 180 + 18 + A.

We are given that (4AB)_6 + (41)_6 = (53A)_6, so we have the equation:

144 + 6A + B + 24 + 1 = 180 + 18 + A

This simplifies to 6A + B = 37.

Since A and B are digits in base 6, they must be between 0 and 5. So, the only possible values for A and B are A = 4 and B = 3.

Therefore, A+B = 4+3 = 7.

Our badminton team has finished 75% of its season.  So far, we have won 20% of the games we played.  What percent of the remainder of our games must we win in order to finish the season with the same number of wins as losses?    

You have won 20% of 75% of the total number of games    

in the season.  That means you've won 15% of the total    

number of games in the season.  

To achieve a 50% winning season, you will need to win   

another 35% of the total number of games in the season.  

However, there are only 25% of the total number of games   

in the season left to play.  There aren't enough games left.    

It's impossible for you to finish the season with 50% wins.    

_.

A = 2 and B = 3, so A + B = 5.

Our badminton team has finished 75% of its season.  So far, we have won 20% of the games we played.  What percent of the remainder of our games must we win in order to finish the season with the same number of wins as losses?    

You have won 20% of 75% of the total number of games    

in the season.  That means you've won 15% of the total    

number of games in the season.  

To achieve a 50% winning season, you will need to win   

another 35% of the total number of games in the season.  

However, there are only 25% of the total number of games   

in the season left to play.  There aren't enough games left.    

Simply stated, you can not achieve a 50% winning season.    

_.

The slope of the line 2x+3y = -5 is -2/3. Since the parallel line has the same slope, the equation of the parallel line will be of the form 2x - 3y = b.

We know that the parallel line passes through the point (-7, 2), so we can plug these values into the equation to solve for b. This gives us:

2 * -7 - 3 * 2 = b -14 - 6 = b -20 = b

Therefore, the equation of the parallel line is 2x - 3y = -20.

The operator that should be used is -.

I have to paint one side of a wall.  The wall is 12 meters tall and 120 meters long.  Each gallon of paint covers 150 square feet. If a foot is approximately 0.3048 meters, then what is the smallest whole number of gallons I can buy and have enough paint to cover the whole wall?  

We're going to have to convert metric units to imperial units,  

so I think it's better to do it from the start.  Even though we'll  

have fractions to contend with, a calculator makes that easy.  

                                                                       12 m  

1 foot is 0.3048 meter, so the height is       –––––––––  =  39.3701 ft   

                                                                   0.3048 m/ft  

                                                                       120 m  

1 foot is 0.3048 meter, so the length is       –––––––––  =  393.7007 ft   

                                                                   0.3048 m/ft  

Area is height times length     (39.3701 ft)(393.7007 ft)  =  15,500.0359 sq ft  

1 gallon will cover 150 sq ft  

so divide that into the area               15,500.0359 sq ft  

                                                        –––––––––––––––  =  103.3335 gallons  

                                                           150 sq ft/gallon  

You can't buy just a third of a gallon of paint,  

so we have to round that up to whole gallons                       104 gallons  

_.

Find the ordered triple (p,q,r) that satisfies the following system:  

p - 2q = 3  

q - 2r = -2 + q  

p + r = 9 + p  

Let's number those equations, for referencing only:  

(1)   p – 2q  =  3   

(2)   q – 2r  =  –2 + q   

(3)   p + r  =  9 + p   

Per equation (2), r = 1   

Per equation (3), r = 9    

You can't have both at the same time.  The system has no solution.  

.

That's the way I was going to do it, when I saw you had already posted this. 

1 in the middle leaves 20.  Run them out to each corner and it's 5 to each corner.   

That makes the square 11 tiles wide, thus the square comprises 121 tiles total.  

121 total, minus the 21 yellow tiles, leaves 100 gray tiles.   

_.

The slope of the line 2x+3y = -5 is 2/3. So, the equation of our parallel line will be of the form y = (2/3)x + b, where b is the y-intercept.

We know that the line passes through the point (-7, 2), so we can plug these values into the equation to solve for b. This gives us:

2 = (2/3)(-7) + b 2 = -14/3 + b 2 + 14/3 = b = 10/3

Therefore, the equation of the parallel line is y = (2/3)x + (10/3). Filling in the blanks, we get:

2x ? + y = 10

The answer is 2x + y = 10.

Actually, the answer is 17/5.  Thanks everyone.