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There are two ways to solve it.

Jenny's grandmother has 24 cats.

Seventeen of the cats do not catch mice.

Ten of the cats have black fur.

What is the smallest possible number of cats that do not catch mice that have black fur?

\(\large{1.} \\ \begin{array}{r|r|r|r} & \text{mice} & \overline{\text{mice}} \\ \hline \text{black fur} & & x & \color{red} 10 \\ \hline \overline{\text{black fur}} & & & \small{24-10} \\ \hline & \small{24-17}& \color{red}17 & \color{red}24 \\ \end{array}\)

\(\large{2.} \\ \begin{array}{r|r|r|r} & \text{mice} & \overline{\text{mice}} \\ \hline \text{black fur} & \small{10-x } & x & 10 \\ \hline \overline{\text{black fur}} & \small{7-(10-x) =} & \small{17-x} & 14 \\ & \small{x-3 } & & \\ \hline & 7 & 17 & 24 \\ \end{array} \)

\(\large{3.} \\ \begin{array}{r|r|r|r} & \text{mice} & \overline{\text{mice}} \\ \hline \text{black fur} & \small{10-x } & x & 10 \\ & \small{\text{min: } 10-x=0 } & & \\ & \Rightarrow x_{min} = \color{red}10 & & \\ \hline \overline{\text{black fur}} & \small{x-3 } & \small{17-x} & 14 \\ & \small{\text{min: } x-3=0} & \small{\text{min: } 17-x=0 } & \\ & \Rightarrow x_{min} = \color{red}3 & \Rightarrow x_{min} = \color{red}17 \\ \hline & 7 & 17 & 24 \\ \end{array}\)

\(\large{4.} \\ \begin{array}{r|r|r|r} & \text{mice} & \overline{\text{mice}} \\ \hline \text{black fur} & \small{10-x } & x & 10 \\ & \Rightarrow x_{min} = \color{red}10 & = min({\color{red}10},{\color{red}3},{\color{red}17}) & \\ & & = 3 & \\ \hline \overline{\text{black fur}} & \small{x-3 } & \small{17-x} & 14 \\ & \Rightarrow x_{min} = \color{red}3 & \Rightarrow x_{min} = \color{red}17 \\ \hline & 7 & 17 & 24 \\ \end{array} \)

\(\large{5.} \\ \begin{array}{r|r|r|r} & \text{mice} & \overline{\text{mice}} \\ \hline \text{black fur} & \small{7 } & \color{red}3 & 10 \\ \hline \overline{\text{black fur}} & \small{0 } & \small{14} & 14 \\ \hline & 7 & 17 & 24 \\ \end{array}\)

1)   The only one that is 10% is the last one   10% of 40 is 4

2) I'd go for the last one there too.

Heureka,  could you please tell me what 

"9 is its own inverse" means 

13.

\(\hphantom{=\,}\ \dfrac{\frac{x}{x+4}}{\frac{1}{x}+\frac{1}{x+4}}\\~\\ =\,\dfrac{\frac{x}{x+4}}{\frac{1}{x}+\frac{1}{x+4}}\cdot\dfrac{x+4}{x+4}\\~\\ =\,\dfrac{x}{\frac{x+4}{x}+1}\\~\\ =\,\dfrac{x}{\frac{x+4}{x}+1}\cdot\dfrac{x}{x}\\~\\ =\,\dfrac{x^2}{x+4+x}\\~\\ =\,\dfrac{x^2}{2x+4}\)

12.

     \( \frac{4-x}{x^2+5x-6}\div\frac{x^2-11x+28}{x^2+7x+6}\)

                                                    Factor the numerators and denominators.

=   \(\frac{-(x-4)}{(x-1)(x+6)}\div\frac{(x-7)(x-4)}{(x+1)(x+6)}\\\)

All the values of  x  that cause a zero in a denominator are:   1,  -6,  7,  4,  -1

So the restrictions are:   x ≠ 1 ,  x ≠ -6 ,  x ≠ 7 ,  x ≠ 4 ,  x ≠ -1

Now let's flip the second fraction and change the sign to multiplication.

=  \(\frac{-(x-4)}{(x-1)(x+6)}\cdot\frac{(x+1)(x+6)}{(x-7)(x-4)}\)

                                                    Multiply the fractions.

=   \(\frac{-(x-4)(x+1)(x+6)}{(x-1)(x+6)(x-7)(x-4)}\)

                                                    Cancel the common factors.

=   \(\frac{-{\color{purple}(x-4)}(x+1){\color{NavyBlue}(x+6)}}{(x-1){\color{NavyBlue}(x+6)}(x-7){\color{purple}(x-4)}}\)

=   \(\frac{-(x+1)}{(x-1)(x-7)}\)

(Hmm...  is   x ≠ -1  not actually a restriction?? Is the expression defined when  x = -1 ?)

Because an inscribed angle is half the angle measure of its intercepted arc,

m∠CAD  =  (1/2)( m arc CD )

m∠CAD  =  (1/2)( 60° )

m∠CAD  =  30°

OA and OC are both radii of circle O, so they are the same length. So △AOC is isosceles, and

m∠OCA  =  m∠CAD 

m∠OCA  =  30°

Since  ∠OBC  and  ∠OBA  form a straight line,

m∠OBC  +  m∠OBA  =  180°

m∠OBC  =  180° - m∠OBA

m∠OBC  =  180° - 60°

m∠OBC  =  120°

Since the sum of the angles in a triangle is  180° ,

m∠BOC  +  m∠OCA  +  m∠OBC  =  180°

m∠BOC  +  30°  +  120°  =  180°

m∠BOC  =  180° - 120° - 30°

m∠BOC  =  30°

Since  m∠BOC  =  m∠OCA ,  triangle OBC  is isosceles, and

the length of BC  =  the length of BO

the length of BC  =  5

The graph  of   y  =x^2   has a vertex  of  (0, 0)

The graph  of   y  = (x + h) - k      shifts this vertex to  the  left  by  h units and down by  k units

So

y  =  ( x + 1)^2  - 2     shifts   (0,0)  to the left 1 unit   and down 2 units.....so.....the "new" vertex  is  (-1,-2)

The graph with this vertex  is the first one

1. B is the correct answer.....the first term, a₁, is  7

And a_n  =  -4*a _n-1

2.   7pi / 6   converted to  degrees is   

7pi / 6  *  180 / pi  =

[ 7pi ]  / [ 6 pi ] * 180  =

(7/6) * 180  =

7 ( 180 / 6 )  =

7 * 30  =

210°.....and this is a 3rd quadrant angle

3.  We can use the tangent here

tan 62  =  x  / 112   multiply both sides by  112

112 * tan 62  = x  ≈  210.6 ft  =  211 ft

4.  None of the answers are correct

cos C  = adjacent side  / hypotenuse  =  15  / 17

These are always a little  difficult

The function (x + 2)^2   is a parabola that opens upward with a vertex of ( -2, 0)

All the graphs show this

The function   l x l  + 1   has a  y value of 

2 at x = -1

1 at x = 0

2 at x  = 1

This eliminates the first three graphs

So....by process of elimination, the last graph  must be the correct one....!!!

Do you see this OK ????

1.  The relative minimum  is the low point on the graph....this is  -36

The relative max  is the high point on the graph....this is 64

2.  f (x) = 8x^2 + 16x  + 6         g (x)  = x^3 - 3x^2  - 9

So   ( f + g) x   =    f (x) + g (x)   =

x^3  + 8x^2 - 3x^2 + 16x + 6  - 9  =

x^3  + 5x^2 + 16x - 3  ......so.....true  !!!

3.  5pi / 3

To convert to  degrees we have

5pi / 3  *    180  / pi  =

5pi / (3 pi) * 180  =

(5/3) * 180  =

5 * (180 / 3)  =

5 * 60  =

300°

The sum is given by

[ distance on day 1 + distance on day 12 ]  *  12  / 2

The distance on day 12 is   

1.5  + .5 (11)   =   7

So...we have

[ 1.5  + 7 ] * 12  / 2  =

[ 8.5 ] * 6  =

51  miles

Your answer is correct....!!!!

Just  enter "6"....then....the multiplication sign, then the square root sign, and then "2"

1.   y= 43(3)^x    would represent the number of pennies after day  x

2. Not sure what this is, but I think we have

f(x)= −0.25(6)^x +2 − 1

The y intercept will be where x  = 0....so we have

-0.25 (6)^{0 + 2}  - 1

-0.25 (6)²  -1

-0.25 (36)  - 1   

-9 - 1  =

-10

3. The practical domain  is  [ 0, 6  ]

Thanks, tertre.....

Here's the "short way"  to find the sum, S₆, at the end of the 6th day....we have that

Sum at any time  =  Initial value  [ 1  - r^n ] / [ 1  - r ]

The initial value is 8

The common ratio, r  = 2

n = the number of days

So we have

S₆  =  8  [ 1 - 2^6]  / [ 1 - 2 ]   =   504  students

Thanks but not the 6th root of 2.  Simply 6 with the square root of 2. Every time I try to enter it on this site's calculator, it puts the 6 under the radical sign instead of out to the left of the sign. I only want the 2 under the radical sign. 

Assuming you already have the diagram, here is what I got:

Since angle EDF is an inscribed angle, we have:

\(\angle{EDF}=\frac{{\text{arc}\ EF}}{2} \)

Since angle A is formed by two tangents, we find:

\(\angle{A}=(\text{arc}\ EDF - \text{arc}\ EF)/2=(360º-2\cdot\text{arc}\ EF)/2\)

Therefore, \(\angle{A}=180º-\text{arc}\ EF = 180º - 2 \angle{EDF}\)

Solving for EDF, we get \(\angle{EDF}=90º-{\angle{A}}/2\)

Since we know angle A is greater than zero, angle EDF is less than 90. Similarly, other angles are also acute. 

I hope this helped,

Gavin

Thanks, tertre....If we want  the sum over all 10 weeks........

First week  =  $4

The common difference is 13

Last week  =  4 + 13(9)  =  $121

So....the sum is

[ First week + Last week ]  *   [ Number of weeks  ] / 2  =

[ 4 + 121 ] *  [ 10 ] / 2  =

125  [ 5 ]   =

$625

f(x)  =  x^4  - 9           g (x)  =  x^3 + 9

So....  (f * g) x  =   f(x) * g(x)  =

[ x^4  - 9 }  *  [ x^3 + 9 ]   =

x^7 + 9x^4  - 9x^3  - 81

Note that the first  few terms are

7,  -28,  112, - 448  ......

So....the first term, a₁,   is   7

And the a_n term is  given  by   -4 (a_n-1)

So....the rule is

a_n  = -4 (a _n-1),   a₁  = 7

We can draw a Venn diagram(sorry, don't have one).

So, 17 of the cats don't catch mice, and 10 have black fur.

Thus, 17+10=27

\(27-24=\boxed{3}\)

The sum will be given by :

100  [ 1  - (1.02)^8  ]  /  [ 1  - 1.02 ]  =

$ 858.30

\(\sqrt[6]{2}\) ?     Use the   \(\sqrt[y]{x}\)    key   

Enter  2

Hit that key

enter 6

enter =