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Catherine rolls a standard 6-sided die six times. If the product of her rolls is  then how many different sequences of rolls could there have been? (The order of the rolls matters.)   

If you go to https://web2.0calc.com/questions/help-plz_51258

you can check out the earlier answer I gave, and GA's elaboration    

on the way to have done it using permutations.  There, I got 45.    

_.

Let the amount of solution A needed==A

Then, the amount of solution B needed==110  -  A

0.55A  +  0.80[110  -  A] ==0.60 * 110, solve for A

A==88 ounces of solution A will be needed

110  -  88 ==22 ounces of solution B will be needed.

Let x be the number of ounces of Solution A and y be the number of ounces of Solution B. We know that x + y = 110 ounces. The total amount of salt in the resulting solution must be 60% of the final solution, so 0.55x + 0.8y = 0.6(x+y) = 0.6 * 110 = 66 ounces. Solving the system of equations x + y = 110 and 0.55x + 0.8y = 66, we get x = 40 and y = 70. Therefore, the scientist should use 40 ounces of Solution A and 70 ounces of Solution B.

Its not 200.

To determine the number of different ways Sam can color the sides of an equilateral triangle using two different colors, we can consider the following cases:

Case 1: All three sides have the same color. In this case, there are two options: either all sides are colored with the first color, or all sides are colored with the second color. So, there are 2 ways to color the triangle in this manner.

Case 2: Two sides have the same color, and the third side is a different color. We need to consider the permutations of colors within this configuration. There are two options for the color of the two sides that are the same, and two options for the color of the third side. So, there are 2 x 2 = 4 ways to color the triangle in this manner.

Case 3: All three sides have different colors. Since Sam has two different colors to choose from, there are two options for each side. So, there are 2 x 2 x 2 = 8 ways to color the triangle in this case.

Therefore, the total number of different ways Sam can color the sides of the equilateral triangle, taking into account the symmetries, is 2 + 4 + 8 = 14.

1  -  2/3 ==1/3

1/3 * 1200 ==400 mL used by Maeve

1200  -  400 ==800 mL left in Maeve's bottle.

1200 mL used by Tanzi

Ratio of Tanzi to Maeve's use ==1200 : 400 ==3 : 1

3 + 1 =4 sum of the ratios.

800 / 4 ==200 - the constant

200  x  3 ==600 mL  Maeve should pour from her bottle into Tanzi's bottle [which is what you are interested in]

200  x  1 ==200 mL should be left in Maeve bottle

Since the ratio of their use is 3 : 1 ==600 : 200 ==3 : 1 when their bottles would be empty at the same time.

Each exterior angle is 1 degree.

The zeros are 0, -9, and -7.

The cannonball is above a height of 6 meters when h(t)≥6. Solving this inequality, we get −4.9t2+14t−5.6≥0. Factoring the left-hand side, we get (−4.9t+7)(t−0.8)≥0. This inequality is satisfied when −4.97​≤t≤0.8. Converting these inequalities to improper fractions, we get 4930​≤t≤98​. Therefore, the cannonball is above a height of 6 meters for a total time of 8​/9 − 30/49​ = 151/441​​ seconds.

I believe it's 13/3.

The quadratic x2+bx+c is positive when its discriminant, b2−4ac, is positive. So we have b2−4ac>0. Expanding the discriminant, we get b2−4ac=(b+2)(b−3)>0. This means that b+2>0 and b−3<0, or b>−2 and b<3. The only value of b that satisfies both of these inequalities is b=2.

Substituting b=2 into the discriminant, we get 4−4c>0, so c<1. The smallest possible value of c is 0, so b+c=2​.

To check our answer, we can plug in b=2 and c=0 into the quadratic and see that it is positive when x<−2 and x>3.

The gum costs 25 cents, and the candy bar costs 1.25 dollars. 

(a)  We can use a system here to solve this. 

Let x=amount of red potion used

Let y=total amount of potion in solution

 
200 + x =  y
0.15(200) + 0.75x = 0.25y

.15(200) because 15% of the blue potion=.15 and there is 200mL. 

.75x because 75% of the red potion=.75 and there is x amount of it 

.25y because we want 25% of the total potion=.25 and there is y amount of it

We can substitute for y in the second equation:
0.15(200) + 0.75x = 0.25(200 + x)

Simplify: 
30 + 0.75x = 50 + 0.25x
-20 = -0.5x
x = 40

Knowing that x=40, we know that y=240, meaning our answer is 240.

b.) We can again use a system for this problem

Let x=Amount of red potion

Let y=Amount of blue potion

x + y = 400
0.3(400) = 0.15y + 0.75x

.3(400)=30% of the 400 mL 

.15y=15% of the magic syrup

.75x=75% of the magic syrup

We can substitute for x in this equation: 
120 = 0.15y + 0.75(400 - y)

Simplify: 

120 = 0.15y + 300 - 0.75y
-180 = -0.6y
y = 300, x = 100

There are 300 mL of blue potion used and 100 mL of red potion used. 

c.) Let x=amount of red potion

Let y=amount of blue potion

0.15x + 0.75y = 0.35(x + y)
.35(x+y) because the total amount of the potion is x+y

Simplify: 

0.15x + 0.75y = 0.35x + 0.35y
0.4y = 0.2x
2y = x

Whenever there is a ratio of 2 (blue potion):1 (red potion), there will be a 35% amount of magical syrup.

For example, 200 mL of blue potion and 100 mL of red potion would work. 

The radius is 2/9.

The error is in answer #1 at   

(A + n )* 7 =  (B - n)     →  7A - 2B  = -5n   (1)   

(A - n) =  3 * ( B + n)  →  A - 3B  = 4n  (2)   

The first one should be  

(A + n)  =  7 • (B – n)    (Since Alice has 7 times as much as Bob, you   

                                      have to multiply Bob's by 7 to make them equal.)   

I do like Guest's reasoning in answer #1 though. 

So let's continue that, but with these other numbers.     

                                         (A + n)  =  7 • (B – n)  

                                          A + n    =  7B – 7n  

                                          A – 7B  =  – 8n            (1)  

                                         (A – n)  =  3 • (B + n)  

                                          A – n   =  3B + 3n 

                                          A – 3B  =  4n                (2)  

Multiply both sides  

of (2) by 2 and then   

add (1) and (2)  

                                            A – 7B  =  – 8n  

                                          2A – 6B   =     8n    

                                          3A – 13B  =  0  

Add 13B to both sides                  3A  = 13B   

Divide both sides by 13B          

                                                     3A        1      

                                                     –––  =  –––   

                                                     13B       1  

Multiply both sides by 13/3   

                                                 13       3A          13        1  

                                                 –––  •  –––   =   –––  •  –––  

                                                   3       13B          3         1   

The digits cancel out   

of the left side, leaving                 

                                                  A         13   

                                                 –––  =  –––   

                                                  B           3   

_.

The quadratic expression x2+bx+c is greater than 0 when its discriminant, b2−4ac, is positive. In this case, b2−4ac>0 only when b2>4ac. This means that b must be positive and c must be negative. Since x2+bx+c>0 for x<−2 and x>3, we know that b<0 and c<0. Therefore, b+c must be negative. The smallest possible value of b+c is −(b+c)=−(−b−c)=b+c=0. Therefore, the value of b+c is 0​.

That answer is wrong.

Solving the quadratic inequality, we get x > -1/2, x < -5/2.  So the inequality is not satisfied for x = -1 and x = -2.  The answer is 2.

https://web2.0calc.com/questions/algebra_89070

Find all values of x such that \sqrt{4x^2} + \sqrt{x^2} = 6 + 3x.   

I thought I had this figured out in an earlier posting of this same question,   

but a moderator, Alan, added an answer, as a correction of the mistake   

that I had made in my answer.  Here's a copy/paste of what Alan said:   

Note there is a difference between  x^2 = 4, for which x = ±2,  and  x = √4, for which x = +2 only.   

√(4x^2) = 2∣x∣    and    √(x^2) = ∣x∣   

so we have:   3∣x∣ = 6 + 3x    

_.

One ordered pair (a,b) satisfies the two equations ab^4 = 48 and ab = 72. What is the value of b in this ordered pair?   

To find b, consider                                                  ab⁴  =  48  

We will divide both sides of that by ab.  

Since ab=72, we will divide the left side  

by "ab" and the right side by its equal 72.  

                                                                              ab⁴         48  

                                                                             ——   =   ——  

                                                                              ab           72  

Note that ab⁴ = (ab) * (b³)  

Cancel ab out of the left side.  

Reduce 48/72 on the right side.  

                                                                               b³           2  

                                                                             ——   =   ——  

                                                                                1            3  

                                                                                 b   =   cube root of (2 / 3)  

_.

Find the coefficient of y^4 in the expansion of (2y-5+3y^2)^6.    

That's way too much multiplication to do by hand.  

Fortunately, a useful tool has been provided for us, at   

https://www.symbolab.com/solver/expand-calculator

Per the "expand" tool on that page, the coefficient of y⁴ is 375.  

But don't take my word for it, go there and see it for yourself.  :-)  

Would it be cheating, to use that website to avoid the drudgery of endless   

multiplications?  I do not consider it cheating to use any tool that has been  

made available.  It used to be slide rules, then calculators, now computers.   

_.