2
A normal die, numbered 1 to 6, is rolled 50 times.
The results are shown in the frequency table.
Score
Frequency
(b) Find the median score.
(a) Write down the modal score.
(c) Calculate the mean score.
(d) The die is then rolled another 10 times.
The mean score for the 60 rolls is 2.95.
Calculate the mean score for the extra 10 rolls.

Answer:

{x = -2, y = -1

Step-by-step explanation:

Solve the following system:

{-3 y + 8 x = -13

5 y - 3 x = 1

Choose an equation and a variable to solve for.

In the first equation, look to solve for x:

{-3 y + 8 x = -13

5 y - 3 x = 1

Isolate terms with x to the left-hand side.

Add 3 y to both sides:

{8 x = 3 y - 13

5 y - 3 x = 1

Hint: | Solve for x.

Divide both sides by 8:

{x = (3 y)/8 - 13/8

5 y - 3 x = 1

Perform a substitution.

Substitute x = (3 y)/8 - 13/8 into the second equation:

{x = (3 y)/8 - 13/8

5 y - 3 ((3 y)/8 - 13/8) = 1

Expand the left-hand side of the equation 5 y - 3 ((3 y)/8 - 13/8) = 1.

5 y - 3 ((3 y)/8 - 13/8) = 5 y + (-(9 y)/8 + 39/8) = (31 y)/8 + 39/8:

{x = (3 y)/8 - 13/8

((31 y)/8 + 39/8) = 1

Choose an equation and a variable to solve for.

In the second equation, look to solve for y:

{x = (3 y)/8 - 13/8

(31 y)/8 + 39/8 = 1

Isolate terms with y to the left-hand side.

Subtract 39/8 from both sides:

{x = (3 y)/8 - 13/8

(31 y)/8 = -31/8

Solve for y.

Multiply both sides by 8/31:

{x = (3 y)/8 - 13/8

y = -1

Hint: | Perform a back substitution.

Substitute y = -1 into the first equation:

Answer: {x = -2, y = -1

The positive intercept y is [tex]\sqrt{ }(3 / 2)[/tex], the negative  y intercept is y= [tex]-\sqrt{ }\left(\frac{3}{2}\right)[/tex], the equation of the tangent line at positive intercept is y=-0.48989x+1.2247,  the equation of the tangent line negative intercept  is y=0.48989x-1.2247.

(a). Here we have to find the positive y intercept

So letting x=0 in the given equation [tex]\frac{1}{(\mathrm{x}-1)^2+\mathrm{y}^2}+\frac{4}{(\mathrm{x}+1)^2+\mathrm{y}^2}=2[/tex]

x=0 implies

[tex]$$\begin{aligned}\frac{1}{(0-1)^2+y^2}+\frac{4}{(0+1)^2+y^2} & =2 \\\frac{1}{1+y^2}+\frac{4}{1+y^2} & =2 \\\frac{5}{1+y^2} & =2 \\5 & =2+2 y^2 \\5-2 & =2 y^2 \\\frac{3}{2} & =y^2 \\\mathrm{y} & =\sqrt{ }\left(\frac{3}{2}\right)\end{aligned}$$[/tex]

Therefore, the positive intercept y is [tex]\sqrt{ }\left(\frac{3}{2}\right)[/tex].

(b). Next we have to find the negative y intercept , the negative y intercept is y= [tex]-\sqrt{ }\left(\frac{3}{2}\right)[/tex]

(c). Next we have to find the equation of the tangent line at [tex]$(0, \sqrt{ }(3 / 2))$[/tex]using implicit differentiation Here

[tex]\frac{1}{(x-1)^2+y^2}+\frac{4}{(x+1)^2+y^2}=2 \\[/tex]

[tex]\left((x-1)^2+y^2\right)^{-1}+4 \times\left((x+1)^2+y^2\right)^{-1}=2[/tex]

Then differentiating using power rule

[tex](-1) \times\left((\mathrm{x}-1)^2+\mathrm{y}^2\right)^{-1-1} \times\left(2(\mathrm{x}-1)+2 \mathrm{y} \times \mathrm{y}^{\prime}\right)+4 \times(-1) \times\left((\mathrm{x}+1)^2+\mathrm{y}^2\right)^{-1-1} \times\left(2(\mathrm{x}+1)+2 \mathrm{yy} \mathrm{y}^{\prime}\right)=0 \\[/tex]

[tex](-1) \times\left((\mathrm{x}-1)^2+\mathrm{y}^2\right)^{-2} \times\left(2(\mathrm{x}-1)+2 \mathrm{y} \times \mathrm{y}^{\prime}\right)+(-4) \times\left((\mathrm{x}+1)^2+\mathrm{y}^2\right)^{-2} \times\left(2(\mathrm{x}+1)+2 \mathrm{yy} \mathrm{y}^{\prime}\right)=0 \\[/tex]

[tex](-1) \times\left((\mathrm{x}-1)^2+\mathrm{y}^2\right)^{-2} \times\left((\mathrm{x}-1)+\mathrm{yy} \mathrm{y}^{\prime}\right)+(-4) \times\left((\mathrm{x}+1)^2+\mathrm{y}^2\right)^{-2} \times((\mathrm{x}+1)+\mathrm{yy})=0[/tex]

Then Substituting x=0 and y= [tex]\sqrt{ }(3 / 2)$[/tex], then

[tex](-1) \times\left(1+\left(\frac{3}{2}\right)\right)^{-2} \times\left(-1+\sqrt{ }\left(\frac{3}{2}\right) \mathrm{y}^{\prime}\right. & =4\left(1+\left(\frac{3}{2}\right)\right)^{-2} \times\left(1+\sqrt{ }\left(\frac{3}{2}\right) \mathrm{y}^{\prime}\right) \\[/tex][tex]\left.-1 \times\left(\frac{2}{5}\right)^2\right) \times\left(-1+\sqrt{ }\left(\frac{3}{2}\right) \mathrm{y}^{\prime}\right) & =4\left(\frac{2}{5}\right)^2 \times\left(1+\sqrt{ }\left(\frac{3}{2}\right) \mathrm{y}^{\prime}\right) \\[/tex]

[tex]-\left(-1+\sqrt{ }\left(\frac{3}{2}\right) \mathrm{y}^{\prime}\right) & =4 \times\left(1+\sqrt{ }\left(\frac{3}{2}\right) \mathrm{y}^{\prime}\right) \\[/tex]

[tex]1-\sqrt{ }\left(\frac{3}{2}\right) \mathrm{y}^{\prime} & =4+4 \sqrt{ }\left(\frac{3}{2}\right) \mathrm{y}^{\prime} \\[/tex][tex]-3 & =5 \sqrt{ }\left(\frac{3}{2}\right) \mathrm{y}^{\prime} \\[/tex]

[tex]\mathrm{y}^{\prime} & =-0.48989[/tex]

Then the equation of the tangent line is y-y1=m(x-x1)

Here X1=0

[tex]& \mathrm{y} 1=\sqrt{ }(3 / 2)=1.2247 \\[/tex]

m=-0.48989

y=-1.2247=-0.48989x

y=-0.48989x+1.2247

(d). Next we have to find the equation of the tangent line at[tex]$(0,-\sqrt{ }(3 / 2))$[/tex] using implicit differentiation

Here [tex]\left((x-1)^2+y^2\right)^{-1}+4 \times\left((x+1)^2+y^2\right)^{-1}=2[/tex]

Then differentiating using power rule

[tex](-1) \times\left((\mathrm{x}-1)^2+\mathrm{y}^2\right)^{-2} \times\left((\mathrm{x}-1)+\mathrm{yy^{ \prime } )}+(-4) \times\left((\mathrm{x}+1)^2+\mathrm{y}^2\right)^{-2} \times\left((\mathrm{x}+1)+\mathrm{y} \mathrm{y}^{\prime}\right)=0\right.[/tex]

Then Substituting x=0 and y=[tex]-\sqrt{ }(3 / 2)$[/tex], then

[tex](-1) \times\left(1+\left(\frac{3}{2}\right)\right)^{-2} \times\left(-1-\sqrt{ }\left(\frac{3}{2}\right) y^{\prime}\right) & =4\left(1+\left(\frac{3}{2}\right)\right)^{-2} \times\left(1-\sqrt{ }\left(\frac{3}{2}\right) y^{\prime}\right) \\[/tex]

[tex](-1) \times\left(-1-\sqrt{ }\left(\frac{3}{2}\right) y^{\prime}\right) & =4 \times\left(1-\sqrt{ }\left(\frac{3}{2}\right) y^{\prime}\right) \\[/tex]

[tex]\left(1+\sqrt{ }\left(\frac{3}{2}\right) y^{\prime}\right) & =4-4 \sqrt{ }\left(\frac{3}{2}\right) y^{\prime} \\[/tex]

[tex]-3 & =-5 \sqrt{ }\left(\frac{3}{2}\right) y^{\prime} \\y^{\prime} & =\frac{3}{5 \sqrt{ }\left(\frac{3}{2}\right)}=0.48989[/tex]

Then the equation of the tangent line is

y+1.2247 =0.48989x

y=0.48989x-1.2247.

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The number of free throws that adam can expect to make if he attempts 20 free throws is 14.

Probability is a metric used to express the possibility or chance that a particular event will occur. Both proportions ranging from 0 to 1 and percentages ranging from 0% to 100% can be used to describe probabilities.

Given the probability that the person making a free throw is 70%. The question asks to estimate how many of his 20 free throw attempts will be successful.

Then, the number of free throws is calculated as follows,

[tex]\begin{aligned}\text{70\% of 20}&=\frac{70\times20}{100}\\&=14\end{aligned}[/tex]

Then, the number of successful free throws is 14.

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Answer:

Team A catches up to B in 2 hours, at the 11 mile mark

Step-by-step explanation:

(5.5 mi/hr)(x) = (4 mi/hr)(x) + 3 mi

(1.5 mi/hr)(x) = 3 mi

x = 3/1.5 = 2 hours

Saturn's distance is 9.5 farther from the Sun than the earth.

An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.

Example: 2 + 3x + 4y = 7 is an expression.

We have,

Earth's distance from the sun = 1.496 x [tex]10^8[/tex] km

Saturn's distance from the sun = 1.4246 x [tex]10^9[/tex] km

Divide Saturn's distance by Earth's distance.

= 1.4246 x [tex]10^9[/tex] / 1.496 x [tex]10^8[/tex]

= 9.5

We see that,

Saturn's distance from the sun x 0.95 = Earth's distance from the sun.

Thus,

Saturn's distance is 0.95 farther from the sun.

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Answer:

Step-by-step explanation:

Enter the value of x below.

x + 9 = 12​

x = 12 - 9

x = 3

-----------------------------------------------

check

3 + 9 = 12

12 = 12

the answer is good

Answer:

1 minute

because the first elevator has traveled 200 meters and the second elevator goes 200 meters in one minute so they will catch up with the first group in 1 minute.

The total cost in dollars of 7 boxes of cookies is; $24.50

From the attached table, we see the following coordinates that represents number of boxes (n) and total cost (c in dollars);

(3, 10.5)

(4, 14)

(5, 17.5)

(9, 31.50)

Now, let us find the slope with two coordinates using the formula;

slope = (y₂ - y₁)/(x₂ - x₁)

Let us use the first two coordinates to get;

Slope = (14 - 10.5)/(4 - 3)

Slope = $3.5

Thus, this is the cost of each box

Thus, for 7 boxes, the cost = 3.5 * 7 = $24.5

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Answer:

The coordinates of B is (4, 7)

Step-by-step explanation:

To find the answer for B, you have to do the inverse of the midpoint formula, which is M equals (x1 multiplied by x2, over 2; y1 multiplied by y2, over 2).

If you substitute in A's coordinates for x1 and y1, you come up with (3, 3) = (2+x, over 2; and -1+y, over 2)

If the x coordinate answer for AB was 3, you can create an equation like this to solve for B's x coordinate: 2+x over 2 equals 3.

         you multiply 2 by 3 to get 6---> 2+x=6-----> x=4

If the y coordinate for AB was also three, you can create another equation to solve for B's y coordinate: -1+y over 2 equals 3.

        you multiply 2 by 3 to get 6 again---->so -1+y=6---->y=7

so the answer for B's coordinates is (4, 7).

Answer:

--------------------------

Given:

Find the coordinates of the other end point B.

Use midpoint equation, considering point B as (x, y):

Therefore the point B has coordinates of (4, - 5).

Answer:

...…...................c = 50

Answer: 12 cups of sugar

Step-by-step explanation:

you can set it up as a ratio

2:4 and this can be simplified to 1:2, so for every cup of flour there is 2 cups of sugar, you could count it by hand but with a ratio of 1:2 you can just multiply by 2 so 6x2=12

Answer:

Step-by-step explanation:

From the truth table, the values for p v (p→q) are all true. Therefore, this proposition is considered a tautology.

A statement or formula that is true under every scenario is referred to as a tautology. For example, the bag is red or it is not. Whatever the color of the bag, these two statements are true.

Construct a truth table for the given propositions. From the table, we can tell that the proposition p v (p→q) is in tautology. Because whatever the value of p and q, the result is true for this proposition.

Therefore, the correct option is option c.

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Answer:

Math Expert Answer

Step-by-step explanation:

52 cans in total separated into x number of bags.  52/x with x being the number of bags.

Answer:

c = 52/x

Step-by-step explanation:

We have 52 total cans and x bags of with an equal amount of cans in each.

Thus, we would have c = 52/x cans per bag. This is probably the answer you are looking for.

However, this can evaluate to a fractional number of cans, so you may be looking to round that answer down to an integer. In that case, the answer would be c = ⌊52/x⌋, which is solving for the greatest integer less than the value of 52/x

π/32 is the area enclosed by the curve r= sin(8θ)

The given curve is polar curve and hence the area of the polar curve is given by:

Let A be the area of the curve so,

A = [tex]\int\limits^a_b {\frac{1}{2} r^2 } \, d\theta[/tex]

where a and b is the boundary at which r=0

so after equation r=0

sin(8θ) =0

=> sin(8θ) =0

=> 8θ = 0,π

=> θ = 0, π/8

so a=0 , b= π/8

now  

   A = [tex]\int\limits^a_b {\frac{1}{2} r^2 } \, d\theta[/tex]    ------(i)

 so   [tex]r^2[/tex] = (sin(8θ))^2

=>  [tex]sin^2[/tex] ( 8θ )

ans we know that

cos(2α) = 1 -  2[tex]sin^2[/tex] α

so   [tex]r^2[/tex]  = (1- cos(16θ) )/2

putting the value of r in the equation (i) we get :-

A = [tex]\int\limits^a_b {\frac{1}{4} *(1-cos(16\alpha ) } \, d\alpha[/tex]

=> 1/4* [tex]\int\limits^a_b {(1-cos(16\alpha ) } \, d\alpha[/tex]

here a=0 and b=π/8

after putting the value and solving the integral

A = π/32

so A is the area enclosed by r=sin(8θ) is π/32

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If the width of a rectangle is two feet less than the length and the perimeter is 52 feet, then the length and width of the rectangle is 14 feet and 12 feet respectively

The width of a rectangle is two feet less than the length

Consider the length of the rectangle as x

Then the width of the rectangle = x - 2

The perimeter of the rectangle = 52 feet

The perimeter of the rectangle = 2(l + w)

Where l is the length and w id the width

Substitute the values in the equation

2(x + x - 2) = 52

2(2x - 2) = 52

2x - 2 = 52/2

2x - 2 = 26

2x = 28

x = 14 feet

Width of the rectangle = 14 - 2

= 12 feet

Therefore, the length and width of the rectangle is 14 feet and 12 feet respectively

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The surface area of a cylinder with a volume of 18 cubic units and a radius of 3 units is 18π +12.

What is the surface area of the cylinder?

The quantity of space enclosing a three-dimensional shape's exterior is its surface area. The area of the cylinder is determined by adding the areas of its two circular bases and curving surface.

Here, we have

Given

The surface area in square units of a cylinder with a volume of 18 cubic units is a function of its radius r in units

where s(r)=2πr² + 36/r ....(1)

Now, we put the value of r = 3 in equation (1) and we get

s(r)=2π(3)² + 36/3

s(r) = 18π + 12

Hence, the surface area of a cylinder with a volume of 18 cubic units and a radius of 3 units is 18π +12.

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Answer:

43.96

Step-by-step explanation:

C = 2πr

C = 2 × 3.14 × 7

C = 43.96

Step-by-step explanation:

given in fig :

radius = 7

so,

circumference of the given figure= 2πr

=2×3.14×7

= 43.96

Answer:

y = 5

Step-by-step explanation:

A line with a slope of 0 is a horizontal line parallel to the x- axis with equation

y = c ( c is the value of the y- coordinates the line passes through )

the line passes through (- 4, 5 ) with y- coordinate 5 , then

y = 5 ← equation of line

Answer:

v=116

u=137

Step-by-step explanation:

hope it's helpful for you

Answer:

55

Step-by-step explanation:

that’s if you meant the interest rate is 3 3/8 for compounded continuously and 3 1/8 for compounded monthly

Answer:

-14/3

Step-by-step explanation:

this is a geometric series with r = -7/8

The infinite sum of geometric series with |r| < 1 is

We already have r, but we need to find a1 which is the first term of the series.

To find it, only plug in 1 for n, 10(-7/8) and this = -35/4

Now plug in r and a1 in the formula above:

[tex]\frac{-35/4}{1+7/8}[/tex]

after simplifying it, you will get the infinite sum = -14/3

Answer: The answers is, 1.77777, repeated 7s

Step-by-step explanation: The explanation if you need to show your work is: Start with the goal of (48 books). Subtract the number of books already read (16 books read).  

(48 - 16) = 32 books left to read.

You must then take the other given of 18 weeks left to complete said goal of (48 books) and divide the number of books left (32 books to read) by the amount of time left (18) weeks to complete the desired goal.

(32 / 18) = 1.77777

Therefore, the answer to your question is that Gracie must read 1.777 books per week to complete her goal on time.    

Good Luck, Hope this helps.

He will make $697.5 in a work week if he sold $4500 worth of merchandise.

Mea Ling earn a weekly salary of $360 which means salary per week is $360.

Then plus a 7.5% commission on sales at a gift shop if  she sold $4,500  worth of merchandise, which means additional.

Divide by 100. Most interest rates are quoted and advertised in terms of a percentage.

7.5%= 0.075

Multiplication In mathematics, multiplication (denoted by ‘ × ’) is a method of finding the product of two or more values.

In arithmetic, multiplication of two numbers represents the repeated addition of one number with respect to another.

0.075×4500= 337.5

Then total she make will be

337.5+360

= $697.5

Therefore, he will make $697.5 in a work week.

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The up front payment costs $34.5 while the payment per miles driven is $0.5

We are told that when renting a vehicle from Schrader car rental, a customer pays a fee up front, and is also charged for miles driven when they return.

Let the amount paid up front be x and let the amount paid per mile driven be y.

Thus, the equation for Henry will be;

x + 85y = 77   -------(1)

The equation for Marcia will be;

x + 115y = 83   ------(2)

Subtract eq 1 from eq 2 to get;

30y = 6

y = 6/30

y = $0.5

Thus;

x + 85(0.5) = 77

x = 77 - 42.5

x = $34.5

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Complete question is;

When renting a vehicle from Schrader car rental, a customer pays a fee up front, and is

also charged for miles driven when they return. Henry drove 85 miles and his rental bill

was $77. Marcia drove 115 miles and her rental bill was $83.

What is the up fronmt cost and the cost per miles driven

Answer: There is one candle for each night of Hanukkah, and the ninth candle, which sits in the middle, is known as the shamash, which translates to attendant or servant candle. This candle is lit first and is used to light the others.

Step-by-step explanation:

The solution to the quadratic equation 3x²-1 = 5x by formula method is

A quadratic equation is a second-order polynomial equation in a single variable of x . It is expressed as ax²+bx+c=0. with a ≠ 0 . Because it is a second-order polynomial equation, the fundamental theorem of algebra guarantees that it has at least one solution. The solution may be real or complex.

In the equation 3x²-1= 5x

we put the equation in standard form. i.e 3x²- 5x -1 = 0

therefore a = 3 , b = -5 and c = -1

formula method is a method of solving the root of a quadratic equation. It is given by x = (-b±√b²-4ac)/2a

=(-(-5±√-5²-4×3×-1)/2×3

=(5±√25+12)/6

=(5±√37)/12

x = 5+6.1/12 or

x= 5-6.1/12

therefore x = 11.1/12 or -1.1/12

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Answer: it's d.

StartFraction 5 + StartRoot 37 EndRoot Over 6 EndFraction, StartFraction 5 minus StartRoot 37 EndRoot Over 6 EndFraction just took the test

The size of the angles is measured in the form of degrees by using a protractor and compass.

The symbol for degrees is a little circle °.

A protractor is a tool that measures the number of degrees in an angle.

To measure an angle with a protractor:

Most protractors have two number scales. It's important to use the same number scale for both sides of the angle.

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The number of scholars who voted for the play named Wicked will be equal to 48.

This problem can be solved by using the concept of Ratio. It is given that the total number of students are 240. Now, the ratio of Annie to Wicked is equal to 5:2 and ratio of Wicked to Wizard Oz is equal to 2:3. If we assume that the number of people who voted for Annie is 'x', the number of people who voted for Wicked is 'y' and number of people who voted for Wizard Oz is 'z'. Now, we write the expressions as

x/y = 5:2; from this we get x = 5y/2                              ....(1)

y/z = 2:3; from this we get z = 3y/2                              ....(2)

x + y + z = 240                                                               ....(3)

We use the above values of equation (1) and (2) in equation (3) as

5y/2 + y + 3y/2 = 240

4y + y = 240

5y = 240

y = 48

So, the number of students who voted for Wicked will be 48.

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Answer:

The heart weighs 404 ounces and a human heart compared to that is 303

Step-by-step explanation:

4 x 102 = 404 - 101

Answer:y=3x+32

Step-by-step explanation:Hope this helps