One edge of a painting is 6 in. longer than the other edge. The painting has a 2-inch-wide frame. The function f(x) = ×2 + 14x + 40 represents the total area of the painting and frame. Find the total area of the painting and the frame if the longer side of the frame is 14 inches long.

The triangles are similar by SAS rule.

Similar triangles are those triangles which have the same shape, but different size.

The corresponding angles of similar triangles are equal and the corresponding sides are proportional.

Given are two triangles ΔGNH and ΔLNM.

∠N is common to both triangles are thus equal.

So ∠N ≅ ∠N

GN / LN = 40 / (12.5 + 40) = 40/52.5 = (16×2.5) / (21×2.5) = 16/21

HN / MN = 32 / (10 + 32) = 32/42 = (16×2) / (21×2) = 16/21

Two corresponding sides are proportional.

So, we have, two corresponding sides are proportional and the corresponding included angles are equal.

This is SAS Similarity rule.

Hence the triangles are similar by SAS similarity rule.

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The rational expression is undefined when the denominator is equal to zero. This is because division by zero is not allowed in mathematics, and the result would be "undefined."

To find the values for which the rational expression is undefined, we need to set the denominator equal to zero and solve for the variable. In this case, the denominator is z:

z = 0

Therefore, the rational expression is undefined when z = 0.

So, the values for which the rational expression ((8x)/(z)) is undefined are:

z = 0

I hope this helps! Let me know if you have any further questions.

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Answer: 4 quarts / 1 gallon

Step-by-step explanation: or C!! hope this helps!!

All the real solutions of the equation are x=3, x=1, and x=-2. The rational root theorem states that any rational root of the equation x³-2x²-5x+6=0 can be written in the form p/q, where p is a factor of the constant term (6) and q is a factor of the leading coefficient (1). Therefore, the possible rational roots of the equation are: ±1, ±2, ±3, ±6

We can use synthetic division to test each of these possible roots until we find one that gives a remainder of 0.

1 | 1 -2 -5 6
 |   1 -1 4
 |_____________
 | 1 -1 -6 10

The remainder is not 0, so 1 is not a root of the equation.

-1 | 1 -2 -5 6
  |  -1  3 2
  |_____________
  | 1 -3 -2 8

The remainder is not 0, so -1 is not a root of the equation.

2 | 1 -2 -5 6
 |   2  0 10
 |_____________
 | 1  0 -5 16

The remainder is not 0, so 2 is not a root of the equation.

-2 | 1 -2 -5 6
  |  -2  8 14
  |_____________
  | 1 -4  3 20

The remainder is not 0, so -2 is not a root of the equation.

3 | 1 -2 -5 6
 |   3  3 0
 |_____________
 | 1  1 -2 6

The remainder is 0, so 3 is a root of the equation. We can use the quotient (1x²+1x-2) to find the remaining roots of the equation.

1x²+1x-2=0

Using the quadratic formula, we can find the remaining roots:

x = (-1 ± √(1²-4(1)(-2)))/(2(1))

x = (-1 ± √(1+8))/2

x = (-1 ± √9)/2

x = (-1 ± 3)/2

x = 1 or x = -2

Therefore, the real solutions using rational root theorem and quadratic formula are of the equation are x=3, x=1, and x=-2.

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The coordinate of P is ( 14/5, 23/5, 28/5)

For the line AB, the direction vectors are

l= 2-1= 1, m=3-1= 2 and n= 4-2= 2

so, the equation of the line AB can be given by

[tex]  \text{$\frac{x-1}{1}$ = $\frac{y-1}{2}$ = $\frac{z-2}{2}$ } [/tex]

let a point be P (x,y,z). lying on the line AB.

coordinate of the general point in AB can be given by

[tex]  \text{$\frac{x-1}{1}$ = $\frac{y-1}{2}$ = $\frac{z-2}{2}$ = t } [/tex]

so, x= t+1, y= 2t+1, z= 2t+2

so, P (x,y,z) can be P (t+1, 2t+1, 2t+2). Another point is C ( 4, 2, 2 ).

the direction ratios of the line CP is

l1= t+1-4 = t-3

m1= 2t+1-2 = 2t-1

n1 = 2t+2-2 = 2t

Dot product of the direction ratios of two orthogonal lines are 0.

so, 1(t-3)+2(2t-1)+2(2t)=0

or, t=9/5

so, x= (9/5)+1 = 14/5

     y= (9/5)*2 + 1= 23/5

     z= (9/5)*2 + 2 = 28/5

so, the coordinate of P is ( 14/5, 23/5, 28/5).

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The solution for \( X \) is \( X = (A - D C)^{-1} D B \).

Assume \( A, B, C, D \), and \( X \) are \( n \times n \) matrices, and assume invertibility of matrices wherever needed. Then, the equation \( A X(B+C X)^{-1}=D \) can be solved for \( X \) by following these steps:

Step 1: Multiply both sides of the equation by \( (B+C X) \) to get rid of the inverse:
\( A X = D(B+C X) \)

Step 2: Distribute the \( D \) on the right side of the equation:
\( A X = D B + D C X \)

Step 3: Rearrange the equation to isolate \( X \) on one side:
\( A X - D C X = D B \)

Step 4: Factor out \( X \) on the left side of the equation:
\( (A - D C) X = D B \)

Step 5: Multiply both sides of the equation by the inverse of \( (A - D C) \) to solve for \( X \):
\( X = (A - D C)^{-1} D B \)

Therefore, the solution for \( X \) is \( X = (A - D C)^{-1} D B \).

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There are 432 possible outcomes when tossing a coin and rolling three number cubes.

Yes, it is possible to calculate the total number of outcomes when tossing a coin and rolling three number cubes using the basic counting method.

According to the basic counting principle, if there are m ways to do one task and n ways to complete another, then there are m * n methods to complete both tasks sequentially.

There are two possible outcomes from this coin toss (heads or tails), and each of the three number cubes can be rolled six different ways (1 through 6). As a result, according to the basic counting principle, there are:

2 * 6 * 6 * 6 = 432

Hence, when rolling three number cubes and tossing a coin, there are 432 possible results.

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The graphs of y = 1/2x - 10 - 3 and y = 1/x are different in shape and behavior, and their only similarity is that they are both continuous functions.

What is a graph ?

Graph can be defined as visual representation of data using ordered pairs.

The graphs of y = 1/2x - 10 - 3 and y = 1/x have different shapes and behaviors.

The graph of y = 1/2x - 10 - 3 is a linear function with a slope of 1/2 and a y-intercept of -13. This means that the graph will have a constant slope of 1/2, and will be downward sloping from left to right. As x increases, y decreases at a steady rate.

On the other hand, the graph of y = 1/x is a non-linear function that is a hyperbola. This means that the graph will be curved and will have two branches, one in the first quadrant and the other in the third quadrant. The graph will approach the x and y-axes but will never touch them. The curve will be symmetric about the line y = x, which means that for every point (x, y) on the graph, there will be a corresponding point (y, x) on the graph.

Therefore, the graphs of y = 1/2x - 10 - 3 and y = 1/x are different in shape and behavior, and their only similarity is that they are both continuous functions.

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Answer:

−4x−3

Step-by-step explanation:

Answer:

Air

Step-by-step explanation:

it's asking you to look at the chart and find which substance is densest by comparing it to how fast sound can move through it (the speed of sound in the substance). In my opinion copper is not the densest material because sound moves through it the fastest; which means the particles in copper are not as densely held together. Air however, shows that sound can only travel 330 meters per second which is incredibly slow compared to the other listed materials, so by that logic it must be the densest.  

Answer:

A - x= -13

Step-by-step explanation:

x + 3 = -10

Step 1) Subtract 3 on both sides

Step 2) x = -13

The degree of the given polynomial is 2 and the leading coefficient is 5. The given expression is 5a^(2)+2a+6. It is a polynomial expression.

The degree of a polynomial is the highest power of the variable in the polynomial. In this case, the highest power of the variable a is 2, so the degree of the polynomial is 2.

The leading coefficient of a polynomial is the coefficient of the term with the highest power of the variable. In this case, the term with the highest power of the variable a is 5a^(2), so the leading coefficient is 5.

Therefore, the degree of the given polynomial is 2 and the leading coefficient is 5.

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The equation of the line passing through the point (8,3) that is parallel to the line y=-3/4+4 is y = (-3/4)x + 9.

The slope of the given line is -3/4. The desired line will likewise have a slope of -3/4 because it is parallel to this line.

The equation of the line travelling through the point (8,3) with a slope of -3/4 may be found using the point-slope form of a line's equation:

y - y1 = m(x - x1)

where the provided point is (x1, y1), and m is the slope.

When we change the values, we obtain:

y - 3 = (-3/4)(x - 8)

Making the right side simpler:

y - 3 = (-3/4)x + 6

To each sides, add 3:

y = (-3/4)x + 9

Hence, y = (-3/4)x + 9 is the equation of the line going through the point (8,3) that is parallel to the line y=-3/4+4.

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The coordinates of the two points that intersect with the y-axis are (2, 3) and (-2, 4)

Represent the points with P and Q

A vertical line is a line whose endpoints have the same x-coordinate

i.e. (x, y1) and (x, y2)

Using the above as a guide, we have the following:

We can make use of the coordinates (x1, y1) and (x2, y2), where the values of x's and y's are not the same

An instance of these points is (2, 3) and (-2, 4)

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Answer:

There are different methods to express a two logarithm into a single logarithm, depending on the specific form of the logarithms involved. Here are some examples:

Logarithms with the same base:

If you have two logarithms with the same base, you can use the following logarithmic identity:

log_a(x) + log_a(y) = log_a(xy)

Using this identity, you can express a sum or difference of logarithms with the same base as a single logarithm:

Example 1: log_2(3) + log_2(5) = log_2(3*5) = log_2(15)

Example 2: log_5(7) - log_5(2) = log_5(7/2)

Logarithms with different bases:

If you have two logarithms with different bases, you can use the following change of base formula to express them with a common base:

log_a(x) = log_b(x) / log_b(a)

Using this formula, you can rewrite a logarithm with a base a as a logarithm with a base b:

Example 1: log_2(7) = log_10(7) / log_10(2)

Example 2: log_5(2) = log_2(2) / log_2(5)

Once you have expressed both logarithms with the same base, you can apply the logarithmic identity from method 1 to simplify them into a single logarithm.

|x| + 2 is always greater than |x| because |x| is always positive or zero.

Absolute value of a number is defined as the positive value of the number. even the number is negative, the absolute value is the positive value of that number.

Consider |x| for any number x.

The possibilities of x are positive number, zero or negative number.

If x is a positive number, |x| is also the same positive number.

2 adding to |x| is also positive number.

So |x| + 2 > |x|

If x is a negative number, |x| is the positive value of that negative number.

So |x| + 2 > |x|.

If x is 0, the |x| = 0

|x| + 2 = 0 + 2 = 2 > |x| = 0

Hence in all cases, |x| + 2 > |x|.

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The ordered pairs are the solution to the inequality as follows:

6) (-1,3) and 7) (1,4).

Inequality is defined as mathematical statements that have a minimum of two terms containing variables or numbers that are not equal.

As per the question, we have:

5) (0,-1)

No, this ordered pair is not a solution to the inequality.

6) (-1,3)

Yes, this ordered pair is a solution to the inequality.

7) (1,4)

Yes, this ordered pair is a solution to the inequality.

8) (0,0)

No, this ordered pair is not a solution to the inequality.

9) (3,3)

No, this ordered pair is not a solution to the inequality.

10) (2, 1)

No, this ordered pair is not a solution to the inequality.

Thus, the ordered pairs are the solution to the inequality as follows:

6) (-1,3) and 7) (1,4).

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Isabel had 10 kilogram of oranges when she started. She ate 2 kilogram of the oranges, leaving her with 8 kilogram. This means that she has 8/10 kilogram of oranges left.

It is important to understand how to calculate fractions in order to answer questions like this. In this case, the question was asking how much of a kilogram remains after Isabel ate 2 kilogram. To answer this, we had to figure out what fraction of the 10 kilogram Isabel ate, which was 2/10. We then subtracted this fraction from 1 in order to figure out how much of a kilogram remains, which was 8/10.

This kind of calculation is useful when trying to figure out how much of something remains after a certain amount has been taken away. It can be used in a wide variety of contexts, such as when trying to figure out how much of a product or item remains after some has been sold or used. Understanding how to calculate fractions can be very helpful in these kinds of scenarios.

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The correct answer is sine and cosine.

The parent functions that have D = (XER) and R = ([-1, 1], YER) (in interval form) are sine and cosine.

Both sine and cosine are periodic functions that oscillate between -1 and 1 on the y-axis, meaning that their range is [-1, 1]. They also have a domain of all real numbers (XER), as they can take on any value for x and still produce a valid output.

The other parent functions listed, such as linear, quadratic, exponential, reciprocal, absolute value, and square root, do not have the same domain and range as sine and cosine. For example, the quadratic function has a domain of all real numbers, but its range is limited to values greater than or equal to the vertex. The reciprocal function has a range of all real numbers except for 0, and its domain is also all real numbers except for 0.

Therefore, the correct answer is sine and cosine.

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Answer:

315

Step-by-step explanation:

The LCM for 45 and 7 is 315

One way to find the LCM of 7 and 45 is to start by comparing the prime factorization of each number. To find the prime factorization, you can follow the instructions for each number here:

What are the Factors of 7?What are the Factors of 45?

Here is the prime factorization of 7:

7171

And this is the prime factorization of 45:

32×5132×51

When you compare the prime factorization of these two numbers, you want to look for the highest power that each prime factor is raised to. In this case, there are these prime factors to consider: 7, 3, 5

The quotient and remainder using (x^(5)-x^(4)+9x^(3)-9x^(2)+9x-12)/(x-1) are 1x^(4)-2x^(3)+11x^(2)-20x+29 and -41, respectively.

To find the quotient and remainder using (x^(5)-x^(4)+9x^(3)-9x^(2)+9x-12)/(x-1), we will use synthetic division.

Write down the coefficients of the dividend polynomial in descending order of the exponent of x. In this case, the coefficients are 1, -1, 9, -9, 9, -12.

Write down the constant term of the divisor polynomial. In this case, the constant term is -1.

Bring down the first coefficient of the dividend polynomial.

Multiply the constant term of the divisor polynomial by the first coefficient of the dividend polynomial and write the result under the second coefficient of the dividend polynomial.

Add the second coefficient of the dividend polynomial and the result from step 4.

Repeat steps 4 and 5 until all the coefficients of the dividend polynomial have been used.

The last result from step 5 is the remainder. The other results are the coefficients of the quotient polynomial.

The synthetic division is shown below:

-1 | 1 -1 9 -9 9 -12
 | -1 2 -11 20 -29
 ---------------------
   1 -2 11 -20 29 -41
 
The quotient is 1x^(4)-2x^(3)+11x^(2)-20x+29 and the remainder is -41.

Therefore, the quotient and remainder using (x^(5)-x^(4)+9x^(3)-9x^(2)+9x-12)/(x-1) are 1x^(4)-2x^(3)+11x^(2)-20x+29 and -41, respectively.

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Answer:

Step-by-step explanation:

The question is illustrated with the attached image.

Angle of elevation is the angle between a line of sight and the horizontal surface.

Let h be the height of the tower

First, calculate distance BC (x).

This is calculated using the following tan ratio,

tan52=h/x

h=xtan52

From the attached image:,

CD=47+x

tan31=h/x+47

h=(x+47)tan31

xtan52=(x+47)tan31,

solving for x we get, x=41.156m

h=xtan52=41.156 * tan52=53.2m

The range of possible side lengths for the third side is 15 yd < x < ∞.

According to the triangle inequality, the lengths of any two sides of any triangle must add up to at least the length of the third side.

We can use the triangle inequality theorem to determine the range of possible side lengths for the third side of each triangle. The theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

10.5 km and 11 km:

Let's call the length of the third side x. Applying the triangle inequality theorem, we get:

5 + x > 11

10 + x > 5

Simplifying, we get:

x > 6

x > -5

Therefore, the range of possible side lengths for the third side is 6 km < x < ∞.

12 mi and 12 mi:

Let's call the length of the third side x. Applying the triangle inequality theorem, we get:

12 + x > 12

12 + x > 12

Simplifying, we get:

x > 0

x > 0

Therefore, the range of possible side lengths for the third side is 0 mi < x < ∞.

25 yd and 10 yd:

Let's call the length of the third side x. Applying the triangle inequality theorem, we get:

10 + x > 25

25 + x > 10

Simplifying, we get:

x > 15

x > -15

Therefore, the range of possible side lengths for the third side is 15 yd < x < ∞.

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The domain of the expression is \( [-1,1] \).

To rewrite the quantity as an algebraic expression of \( x \), we need to use the definition of the cotangent function and the relationship between the sine and cosine functions. The cotangent function is defined as \[ \cot (\theta)=\frac{1}{\tan (\theta)}=\frac{\cos (\theta)}{\sin (\theta)} \]Using the relationship between the sine and cosine functions, \[ \cos^{2} (\theta)=1-\sin^{2} (\theta) \]we can rewrite the cotangent function in terms of the sine function as \[ \cot (\theta)=\frac{\sqrt{1-\sin^{2} (\theta)}}{\sin (\theta)} \]Substituting \( \arcsin (x) \) for \( \theta \) gives us \[ \cot (\arcsin (x))=\frac{\sqrt{1-\sin^{2} (\arcsin (x))}}{\sin (\arcsin (x))} \]Since \( \sin (\arcsin (x))=x \), we can simplify the expression to get \[ \cot (\arcsin (x))=\frac{\sqrt{1-x^{2}}}{x} \]This is the algebraic expression of the quantity in terms of \( x \).

The domain of the expression is the set of values of \( x \) for which the expression is defined. Since the expression involves a square root, we need to ensure that the radicand is non-negative. This gives us the inequality \[ 1-x^{2}\geq 0 \]Solving for \( x \) gives us the inequality \[ -1\leq x\leq 1 \]Therefore, the domain of the expression is \( [-1,1] \).

In conclusion, the quantity can be rewritten as an algebraic expression of \( x \) as \[ \cot (\arcsin (x))=\frac{\sqrt{1-x^{2}}}{x} \]and the domain on which the equivalence is valid is \( [-1,1] \).

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I. Simplify the following fully:

1.1 (2x-1)(x^(2)-3x+1)

= 2x^(3) - 6x^(2) + 2x - x^(2) + 3x - 1

= 2x^(3) - 7x^(2) + 5x - 1

1.2 12(2x+3)(5-x)

= 12(-2x^(2) + 7x + 15)

= -24x^(2) + 84x + 180

1.3 (x^(3)-1)/((x+2)+x(x+2))+(x-1)/(2x+4)

= (x^(3)-1)/(x^(2)+3x+2)+(x-1)/(2(x+2))

= (x-1)(x^(2)+x+1)/(x+2)(x+1)+(x-1)/(2(x+2))

= (2(x-1)(x^(2)+x+1)+(x-1)(x+1))/(2(x+2)(x+1))

= (2x^(3)+x^(2)-x-1+x^(2)-1)/(2(x+2)(x+1))

= (2x^(3)+2x^(2)-x-2)/(2(x+2)(x+1))

1.4 (3)/(x^(2)-9)+(2)/((x-3)^(2))

= (3)/((x+3)(x-3))+(2)/((x-3)^(2))

= (3(x-3)+2(x+3))/(x+3)(x-3)^(2)

= (3x-9+2x+6)/(x+3)(x-3)^(2)

= (5x-3)/(x+3)(x-3)^(2)

2. Factorise:

2.1 6x^(2)+7x-20

= (2x-4)(3x+5)

2.2 x^(3)+x^(2)-x-1

= (x+1)(x^(2)-1)

= (x+1)(x+1)(x-1)

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Answer:

0.866

Step-by-step explanation:

sin (200) = −0.34202014

cos (80) = 0.17364817

-

cos (200) = −0.93969262

sin (80) = 0.98480775

=

0.86602540338, or 0.866

Martin buys a total of 2(5)/(6) + 4(3)/(4) = 10/6 + 19/4 = 40/24 + 57/24 = 97/24 pounds of nuts.

To find the total amount of nuts Martin buys, we need to add the amount of peanuts and the amount of almonds he buys.

First, we need to convert the mixed numbers to improper fractions. To do this, we multiply the whole number by the denominator and add the numerator. For 2(5)/(6), we multiply 2 by 6 and add 5 to get 10/6. For 4(3)/(4), we multiply 4 by 4 and add 3 to get 19/4.

Next, we need to find a common denominator in order to add the two fractions. The least common denominator for 6 and 4 is 24, so we multiply the numerator and denominator of 10/6 by 4 to get 40/24 and the numerator and denominator of 19/4 by 6 to get 57/24.

Finally, we add the two fractions by adding the numerators and keeping the same denominator: 40/24 + 57/24 = 97/24.

Therefore, Martin buys a total of 97/24 pounds of nuts.

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The quotient is -6x^2-2x-2 and the remainder is -4.

To find the quotient and remainder for the given expression, we can use long division.

First, we divide the first term of the dividend (-6x^3) by the first term of the divisor (x) to get -6x^2. This is the first term of the quotient.

Next, we multiply the first term of the quotient (-6x^2) by the divisor (x-1) to get -6x^3+6x^2. We subtract this from the dividend to get -2x^2-2.

We repeat this process with the new dividend (-2x^2-2) and the same divisor (x-1). We divide the first term of the new dividend (-2x^2) by the first term of the divisor (x) to get -2x. This is the second term of the quotient.

We multiply the second term of the quotient (-2x) by the divisor (x-1) to get -2x^2+2x. We subtract this from the new dividend to get -2x-2.

We repeat this process one more time with the new dividend (-2x-2) and the same divisor (x-1). We divide the first term of the new dividend (-2x) by the first term of the divisor (x) to get -2. This is the third term of the quotient.

We multiply the third term of the quotient (-2) by the divisor (x-1) to get -2x+2. We subtract this from the new dividend to get -4. This is the remainder.

So, the quotient is -6x^2-2x-2 and the remainder is -4.

We can check our work by verifying that

(Quotient)(Divisor) + Remainder = Dividend:

(-6x^2-2x-2)(x-1) + (-4) = -6x^3+6x^2-2x^2+2x+2x-2-4 = -6x^3+4x^2-2

Therefore, our answer is correct.

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Answer:

Step-by-step explanation:

you have 22 bucks and your mom took 2

22-2=20

My Answer:

22-2= 20, right? She can't drink anymore wine if you (shown as "i") used the remaining 20$. Tell me if this makes sense. No wine (none) is consumed if you have no money left.

Answer from Artificial Intellegence (to get another view on the question so you can figure out the best answer):

If you had 22 bucks and your mom took 2 dollars for her weekly wine tasting, you would have 20 dollars left. If you have 0 dollars left, this means your mom spent all of the remaining 20 dollars on wine. Assuming each wine bottle costs the same, we can divide the remaining 20 dollars by the cost of one wine bottle to find how many wine bottles your mom bought for her weekly wine tasting. Without knowing the exact cost of one wine bottle, we can't determine the exact number of wine bottles your mom drank.