Find a basis for the space of 2x2 diagonal matrices:

Answer:

[tex]y=\frac{2}{3}x+4[/tex]

Step-by-step explanation:

The line passes through [tex](-3,2)[/tex] and [tex](3,6)[/tex], and so has a slope of [tex]\frac{6-2}{3-(-3)}=\frac{2}{3}[/tex].

Since the line has a [tex]y[/tex]-intercept of [tex]4[/tex], the equaton is [tex]y=\frac{2}{3}x+4[/tex].

Answer:

Its option A I believe (its closest)

Step-by-step explanation:

I used Desmos (the graphing site), and it was the closest answer.

The fourth term of the resulting four-term sequence is 206.

Let the four terms of the arithmetic sequence be a a+d, a+2d, and a+3d (where d is common difference)

Let the four terms of the geometric sequence be b, br, br^2, and br^3 (where r is the common ratio)

An arithmetic sequence is a sequence of numbers where the difference between any two consecutive terms is always the same. The difference is called the common difference.

The resulting four-term sequence is a+b, a+d+br, a+2d+br^2, a+3d+br^3

Given that the first three terms of the resulting four-term sequence are 57, 60, and 91, we can set up the following equations:

a + b = 57

a + d + br = 60

a + 2d + br^2 = 91

By substituting the first equation into the second and third equations we get:

d + b(r-1) = 60 - 57 = 3

2d + b(r-1)^2 = 91 - 57 = 34

By substituting the first equation and the second equation into the third one we get:

2(3) + b(r-1)^2 = 34

6+b(r-1)^2 = 34

b(r-1)^2 = 28

Note that either b=28 or b=7. We proceed with casework:

If b=28, then r=2,a=29, and d=25.The arithmetic sequence is 29,4,-21,-46, arriving at a contradiction.

If b=7, then r=3,a=50, and d=-11.The arithmetic sequence is 50,39,28,17, and the geometric sequence is 7,21,63,189. This case is valid.

Therefore, The fourth term of the resulting four-term sequence is [tex]a+3d^{2}+br^{3}= 17 +189 =206[/tex].

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Answer:

To solve this problem, we will use the following steps:

Step 1: Nick and Chloe paddled downstream at an average speed of 12 km/h and back upstream at an average rate of 8 km/h. So, we can find the total distance they travelled by finding the distance they travelled downstream plus the distance they travelled upstream.

Step 2: We know that the total trip took 7 hours and they took a 45-minute lunch break. To convert 45 minutes to hours, we can divide 45 by 60, so 45/60 = 0.75 hours.

Step 3: We can then subtract the time they took for lunch break from the total trip time, so 7 - 0.75 = 6.25 hours

Step 4: To find the distance travelled downstream, we can multiply the time spent traveling downstream by the speed of the canoe, so 6.25 x 12 = 75 km

Step 5: To find the distance travelled upstream, we can multiply the time spent traveling upstream by the speed of the canoe, so 6.25 x 8 = 50 km

Step 6: We can then add the distance travelled downstream and the distance travelled upstream to find the total distance travelled, so 75 + 50 = 125 km

Final Answer: Nick and Chloe travelled a total distance of 125 km.

The measurement of length of the car is 2.25m, when the width is 1.5 meter. Thus, option B is correct.

Measurement is the process of relating numerical values to physical quantities and phenomena. The sciences, engineering, construction, and other technical fields, as well as nearly all daily activities, all depend on measurement. The components, circumstances, restrictions, and theoretical underpinnings of measurement have thus been extensively studied.

Measurements can be made using only the human senses, in which case they are frequently referred to as estimates, or, more frequently, by using instruments.

These instruments can range in complexity from straightforward rules for measuring lengths to extremely complex systems built to detect and measure things that are completely outside the range of the senses, like radio waves from a far-off star or the magnetic moment of a subatomic particle.

Given that length : width is  3 : 2

let length be 3x and width be 2x

width is given 1.5m

2x = 1.5

x = 1.5/2

x = 3/4

Length = 3x

= 3 × 3/4

= 9/4

= 2.25 m

Thus, the measurement of length of the car is 2.25m, when the width is 1.5 meter.

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Correct answer:

The flat dimensions of a car is given as 3 : 2, and the width of the car 1.5 metre. which measurement could be the length of a car?

a. 2m

b. 2.25m

c. 2.50m

d. 2.15m

Answer:

1000 mitre=1km

1.25 ×1000

Answer:

DIGIT FORM = 13km

DECIMAL FORM: 12.5km

Step-by-step explanation:

DIGIT Form: A digit can only be a whole number, so round 12 and a half (12/5 or 12.5) to the nearest whole number.

DECIMAL Form: It cannot be a whole number and must consist of a number followed by a decimal point (a period) and then another number.

DON'T FORGET TO ADD KILOMETERS (km) AFTER YOUR ANSWER.

The dimensions of the rectangular plot that require the least amount of fencing is :

A(17.32, 17.32) = 600

P(17.32, 17.32) = 121.24

Now, According to the question:

600 [tex]u^2[/tex] of land to be enclosed by fence

Area inside plot should be maximized, amount of fence should be minimized

If we draw a quick sketch, we get something like this:

_____ _____

|        |         |

|        |         |

|        |         |

|        |         |

The two rectangles could be different sizes but if one rectangle size maximizes area then it would make sense to use that some rectangle for both sides.

Let's put some variables into our picture so we can define our constraints.

       y        y  

  |         |        |

x |      x |      x |

  |         |        |

  |         |        |

Now we just need an equation to connect our unknowns (how much fence to use) to our constraint (it has to have an exact area of 600 u^2.

A = x × 2y = area = 600

P = 3x + 4y = perimeter = a minimum

First we solve the area equation for y:

y = [tex]\frac{600}{2x}[/tex] = [tex]\frac{300}{x}[/tex]  

and then we substitute that result into our perimeter equation:

P = 3x + 4 (300/x)

P = 3x + 1200/x

Our perimeter equation is: P(x) = 3x + 1200/x OR P(x) = 3x +1200[tex]x^-^1[/tex]

We can find what value of x makes our perimeter as small as possible (a minimum) by graphing or by using Calculus.

In Calculus we know that maximums and minimums always occur when the derivative is zero. This is because the slope (rate of change) at these points in a graph are always flat or zero. Our derivative is

P'(x) = 3 - 1200[tex]x^-^2[/tex]

When set equal to zero, we can solve for x:

3 - 1200[tex]x^-^2[/tex]= 0

1 - 400[tex]x^-^2[/tex] = 0

1 = 400/[tex]x^{2}[/tex]

[tex]x^{2}[/tex] = 400

x = ± 20

Our fence length can't be a negative number so our answer must be 20u.

If our x length is 20, then our y length must be 300/x to ensure our area is 600 u^2.

The y length is 300/(20) = 15

Let's put all of our numbers into our picture to see if it make sense.

      15      15    

    |        |         |

20 |    20 |    20 |

    |        |         |

    |        |         |

       15      15

Sure enough our area is 600.

Here's what we have right now:

A(20, 15) = 600

P(20, 15) = 120

How about we try something a little bigger like 20.5 for x and 14.63 for y?

A(20.5, 14.63) = 600

P(20.5, 14.63) = 120.02

How about we try something a little smaller like 19.5 for x and 15.38 for y?

A(19.5, 15.38) = 600

P(19.5, 15.38) = 120.02

It looks like if we try to change x, even a little, to make it bigger or smaller the perimeter only gets bigger.

You might have guessed at the beginning that a square is the best shape to use to maximize area, so let's try that as well.

Each rectangle is 300 u^2. We want a square so each side would be √(300) = 17.32

              17.32         17.32    

        |                |                 |

17.32 |       17.32 |        17.32 |

        |                |                 |

             17.32          17.32

A(17.32, 17.32) = 600

P(17.32, 17.32) = 121.24

Hence, The dimensions of the rectangular plot that require the least amount of fencing is :

A(17.32, 17.32) = 600

P(17.32, 17.32) = 121.24

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As per the concept of estimation, Len's estimate and what is the actual sum of the numbers is 342

The rule for decimal in math is known as if the number after the decimal point is greater than or equal to 5 then we have to add one number to the whole number and remove the decimal terms.

Where as if the number after the decimal point is less than 5 then we have to simply remove the decimal term and the whole number is as it is the same.

So, here we have the nearest whole of 16.23 is written as 16 because here the decimal is less than 5, then it is same whole.

And the nearest whole of 215.7 is equal to 216 because here decimal is greater than 5, so we have to add 1 number

Now, the nearest whole of 110.3 is equal to 110 where as the decimal is less than 5 then it is same whole

Then the Estimated sum  is calculated as,

=> 16 + 216 + 110

=> 342

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If the number y is divided by x, the quotient is p and the remainder is r , then the relation between x,y,p,r is given by the expression [tex]y=xq+r[/tex]  .

What is Euclid's Division Lemma ?

The Euclid's Division Lemma states that for two positive integers , a and b , then there exist unique integers q and r such that : [tex]a=bq+r,\; \; where \; 0\leq r < b[/tex] ;

here , the number y is divided by x , that means , the divisor is "x" , the dividend is "y" ,

the quotient is denoted by "q" ;

and the remainder is denoted by  "r" ;

So , According to Division Lemma :

we can write ; the relation between x,y,p,r as ⇒  [tex]y=xq+r[/tex] , where [tex]where \; 0\leq r < x[/tex] .

Therefore , the relation between  x,y,p,r is [tex]y=xq+r[/tex] .

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Answer:

  251.5 square units

Step-by-step explanation:

You want to know the area of the given pentagonal figure.

The figure can be decomposed into a (top) triangle that is 21 units wide and (19-8) = 11 units high, along with a (bottom) trapezoid that is 8 units high and has bases of 21 units and 13 units. The area of the figure is the sum of the areas of these parts.

The area of the triangle is ...

  A = 1/2bh

  A = 1/2(21)(11) = 115.5 . . . . square units

The area of the trapezoid is ...

  A = 1/2(b1 +b2)h

  A = 1/2(21 +13)(8) = 136 . . . . square units

The area of the pentagon shown is the sum of these areas:

  115.5 +136 = 251.5 . . . . square units

The equation of the normal line of the curve f(x) = x³ - 14x² + 64x - 93 at point (3,0) is y = (-1/7) x +3/7

The slope of a tangent line of a curve at a given point is equal to its derivative at that point.

The normal line is a line which perpendicular to tangent line. Two lines are perpendicular to each other if the product of their slopes is equal to -1.

The equation of the curve is:

   f(x) = x³ - 14x² + 64x - 93

Take the derivative:

f '(x) = 3x² - 28x + 64

Therefore, the slope of the tangent line at point (3,0) is:

f '(3) = 3(3)² - 28(3) + 64 = 7

Hence, the slope of the normal line is:

m = -1/7

The equation of the normal line:

(y - 0) = (-1/7) (x - 3)

y = (-1/7) x +3/7

Your question is incomplete. Most likely, it was:

For each problem, find the equation of the line normal to the function at the given point. If the normal line is a vertical line, indicate so. Otherwise, your answer should be in slope-intercept form.

   f(x) = x³ - 14x² + 64x - 93 at (3,0)

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The surface area of the prism shown in the figure is equal to 480 square inches. (Right choice: B)

In this problem we need to determine the surface area of a prism, that is, the sum of the areas of the five faces of the prism. There are three rectangles and two right triangles as faces of the prism, whose area formulas are described below:

Rectangle

A = w · l

Triangle

A = 0.5 · w · l

Where:

Now we proceed to determine the areas of all faces of the prism:

A = (8 in) · (9 in) + 2 · 0.5 · (15 in) · (8 in) + (9 in) · (15 in) + (17 in) · (9 in)

A = 480 in²

The prism shown in the figure has a surface area of 480 square inches.

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The method proposed is not fair, it can be adjusted by giving each sibling equal chance of winning.

What is algebra ?

Algebra is a branch of mathematics that deals with mathematical symbols and the rules for manipulating these symbols. It includes the study of operations such as addition, subtraction, multiplication, division, and the extraction of roots, as well as the properties of operations such as commutativity, associativity, and distributivity.

The method proposed by Alan is not fair because it does not take into account the probability of each outcome. The product of the outcomes being even or odd does not guarantee an equal probability of winning for both siblings. To make it fair, each sibling should have an equal chance of winning. A fair method could be to have the arrow spun once, and the sibling whose number is spun gets to choose the new pet. Another fair method could be to have a coin flipped, and the sibling whose side of the coin is facing up gets to choose the new pet.

So , The method proposed is not fair, it can be adjusted by giving each sibling equal chance of winning.

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The amount of wood, in meters, needed to rebuild the frame is 1.8288 m.

What is amount and example?

amount to (something) : to produce (a total) when added together. The bill amounted to 10 dollars. They have debts amounting to thousands of dollars.

To calculate the amount of wood, in meters, needed to rebuild the frame, we can use the following formula:

wood (m) = (wood (ft) x 0.3048) / 6

Since each piece of wood provided measures 6 ft (approximately 1.8 to 8.7), we can calculate the amount of wood, in meters, needed to rebuild the frame as follows:

wood (m) = (6 ft x 0.3048) / 6

wood (m) = 1.8288 m      

Therefore, the amount of wood, in meters, needed to rebuild the frame is 1.8288 m.

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Answer: 22.9

Step-by-step explanation:

1. 40 divided by 1.75 = 22.8571428571

2. Round that up to 22.9

Answer:

You don't need to overthink this one, it's not asking for any exact values or estimates so all you do is 40/1.75 which gives you 22.8571428571. Since it's asking for 1 decimal place then the answer would be 22.9 litres of petrol.

The quadratic equation y = x² - 2x - 3 have ordered points at (0, -3), (2, -3), (-2, 5) and (4, 5)

An equation is an expression composed of variables and numbers linked together by mathematical operations.

Quadratic equation is a polynomial that have a degree of 2. The degree of a polynomial is the highest exponent of the variables.

Given that:

y = x² - 2x - 3

When x = 0; y = (0)² - 2(0) - 3 = -3

When x = 2; y = (2)² - 2(2) - 3 = -3

When x = -2; y = (-2)² - 2(-2) - 3 = 5

When x = 4; y = (4)² - 2(4) - 3 = 5

The equation y = x² - 2x - 3 is a quadratic equation.

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we are looking for a point to the right of (4,7), the point P is (15/2, x+3) = (15/2, 21/2).

What is polynomial ?

A polynomial is a mathematical expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

Let P = (x, y) be a point on L that is four times as far from (2, 5) as it is from (4, 7). Since L contains P, we know that y = x + 3. We also know that the distance from P to (2, 5) is 4 times the distance from P to (4, 7). Using the distance formula, we get:

(x - 2)^2 + (y - 5)^2 = 4[(x - 4)^2 + (y - 7)^2]

Plug in y = x+3 and simplify the equation as ,

(x-2)^2 + (x+3-5)^2 = 4(x-4)^2 + 4(x+3-7)^2

x^2 - 4x + 4 + x^2 - 6x + 9 = 4x^2 - 32x + 48 + 4x^2 - 24x + 36

2x^2 - 30x + 97 = 0

Solving for x, we have x = 15/2 or x = -3/1

Since we are looking for a point to the right of (4,7), the point P is (15/2, x+3) = (15/2, 21/2).

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Answer: In five days, there will be around 2,476 bacteria.

Step-by-step explanation: Using the hint given, simply plug in the numbers to calculate the predicted amount of bacteria. Hope this helps!

[tex]100(1+0.9)^{5}=2,476.099[/tex]

Answer:

2476

Step-by-step explanation:

Given formula:

[tex]\textsf{Future Amount}=\sf I(1+r)^t[/tex]

Given values:

Substitute the given values into the given formula and solve:

[tex]\begin{aligned}\implies \textsf{Future Amount}& =\sf 100(1+0.9)^5\\& =\sf 100(1.9)^5\\& =\sf 100(24.76099)\\& =\sf 2476.099\\& =\sf 2476\;(nearest\;whole\;number)\end{aligned}[/tex]

Answer:

Step-by-step explanation:

35% of 230 can be found by first converting 35% to a decimal. To do this, divide 35 by 100.

35/100 = 0.35

Next, multiply the decimal by the total number you want to find the percentage of, which is 230 in this case.

0.35 * 230 = 80.5

So 35% of 230 is 80.5.

Answer:

  see attached

Step-by-step explanation:

You want the steps in order for constructing a perpendicular to a line through an external point.

Essentially, the procedure identifies two points on the line (P and Q) that are equidistant from the external point (R), then proceeds to draw a perpendicular bisector of the segment between those two points. The external point is already equidistant from P and Q, so we only need to find another equidistant point (S) to complete the construction.

See the attachment for the ordering of the instruction steps. (Note the step numbers are not in order on the left.)

Leading coefficient: 23

Degree: 8 (The degree of a polynomial is the highest power of the variable in the polynomial. In this case, the highest power of the variable "u" is 8, so the degree of the polynomial is 8.)

Answer:

i) 23

ii) 8

Step-by-step explanation:

i) leading coefficient:

23

ii) Degree

The degree of the polynomial is the highest power in the polynomial.

So,

degree of polynomial=8

1/0 is undefined but not infinity.

In mathematics, expressions like 1/0 are undefined. But the limit of the expression 1/x as x tends to zero is infinity.

                          i.e limx→0 1/x = infinity

Similarly, expressions like 0/0 are undefined. But the limit of some expressions may take such forms when the variable takes a certain value and these are called indeterminate. limit tends to some value of x varies with the expression so as there is limit of some value i.e x but not a particular limit 0f 1/x as x tends to zero is infinity. But 1/0 is not infinity and 0/0 is not indeterminate, since division by zero is not defined.

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The complete question is:

Is 1/0  infinty or undefined?

The radius of the signal is 10 units. The 1st option is the answer

The equation of a circle is a mathematical expression that describes the properties of a circle. It is typically represented in the form of (x - h)²+ (y - k)² = r², where (h, k) is the center of the circle, and r is the radius

Since the border is modeled by the equation (x – 10)² + y²= r² and the boundary is located at (2, 6).

To get the radius of the signal substitute x = 2 and y = 6 into the given equation and solve for r:

(x – 10)² + y²= r²

(2 – 10)² + 6²= r²

(-8)² + 6²= r²

64 + 36 =  r²

100 =  r²

r² = 100

r = √100

r = 10 units

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Given: f(d) = 11(1.01)d, which will display equation of the radius of the algae.

Simple Equation: What Is It? two phrases along either side of a sign are connected by a mathematical equation. Usually, it only includes one variable and an equal sign. Example: 2x – 4 = 2.

Solution:

Part A: 11.79 = 11(1.01)d(1.01)d(1.01)d(1.01)d(1.01)d= 11.79/11 (1.01)d(1.01)d(1.01)d(1.01)d(1.01)d(1.01)d(1.01)d(1.01)d(1.01)d(1.01)d(1.01)d(1.01)d(1.01)d(1.01)d(1.01)d(1.01)d= 1.071818 d = log 1.0718

When domain = 0 d = 0 f(0) = 11, the y-intercept is found (Part B) (1.01)

^{0}11(1.01)0\s= 11 x 1 = 11 f(d) = 11

Part C: Average rate of change, f/d = (f(7) - f(2)) / (7 - 2) = (11(1.01)711(1.01)211(1.01) 2) / 5 = 0.57 / 5 = 0.114

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Given: f(d) = 11(1.01)d, which will display equation of the radius of the algae.

Simple Equation: What Is It? two phrases along either side of a sign are connected by a mathematical equation. Usually, it only includes one variable and an equal sign. Example: 2x – 4 = 2.

Solution:

Given:

f(d) = 11(1.01)d

To Find:

Part A: When the biologist concluded her study, the radius of the algae was approximately 11.79 mm. What is a reasonable domain to plot the growth function?

Part B: What does the y-intercept of the graph of the function f(d) represent?

Part A:

[tex]11.79 = 11(1.01)^{d}11(1.01)d\\(1.01)^{d}(1.01)d\\= 11.79/11(1.01)^{d}(1.01)\\= 1.071818\\d = log 1.071818 / log 1.01\\d = 6.97[/tex]

Part B:

y-intercept is obtained when domain = 0

[tex]d = 0 ⇒ f(0) = 11(1.01)^{0}11(1.01)0\\= 11 x 1 = 11\\f(d) = 11[/tex]

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On solving the provide question, we can say that by quadratic equation  number of corn plants in the row are 52 - 4 = 21

A quadratic equation is x ax2+bx+c=0, which is a quadratic polynomial in a single variable. a 0. The Fundamental Theorem of Algebra ensures that it has at least one solution since this polynomial is of second order. Solutions may be simple or complicated. An equation that is quadratic is a quadratic equation. This indicates that it has at least one word that has to be squared. The formula "ax2 + bx + c = 0" is one of the often used solutions for quadratic equations. where are numerical coefficients or constants a, b, and c. where the variable "X" is unidentified.

number of rows be x.

the number of corn plants = x - 4

So,

[tex]x(x - 4) = 525\\x^2 - 4x - 525 = 0\\x = 25, x = -21\\[/tex]

number of corn plants in the row are 52 - 4 = 21

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True. A function whose values repeat based on positions of a point that moves around a circle is called a sinusoid, using trigonometry approach.

The sine waves oscillate in a smooth, repetitive manner, defined in terms of distance from the [tex]x[/tex] axis to the point on the unit circle.

The main features of a sinusoidal function are its amplitude, its phase, and its frequency. Let us discuss the features one by one :

Amplitude specifies the maximum distance between the x axis i.e. the horizontal axis and the vertical position of the point or a the waveform. It is denoted by [tex]A[/tex].

Phase of a function refers to the horizontal position of the waveform with correspondence to one rotation. The extent or angle by which, a point or a waveform is shifted, is called the phase difference or phase shift.

Frequency is the measurement which tells us about how quickly the sinusoid completes cycles i.e. it simplifies the movement of the sinusoidal wave per unit time.

The sinusoid functions play key role in modern periodic phenomena such as sound and light waves, oscillators, average temperature variations throughout the year, sunlight intensity ,etc.

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Answer:

Step-by-step explanation:

The value of X is 45 because all the angles of a triangle add up to 180 degrees. The triangle is a right triangle as shown in the picture, the square in the corner of the triangle indicates this, any right angle is 90 degrees and when you minus 90 from 180 you get 90. Now, there are two more angles in the triangle so the value of X is 45, (90/2=45).

Answer:

The capacity of the pot can be calculated by finding the volume of the clay pot. We can use the formula for the volume of a sphere to find the volume of the inside and outside of the pot and subtract the smaller volume from the larger volume to find the volume of the clay.

Step-by-step explanation:

The formula for the volume of a sphere is:

V = (4/3)πr^3

Where V is the volume and r is the radius.

First, we will calculate the volume of the inside of the pot:

V_inner = (4/3)π(8.2cm)^3 = (4/3)π(530.24cm^3) = 707.68π cm^3

Then, we will calculate the volume of the outside of the pot:

V_outer = (4/3)π(9.1cm)^3 = (4/3)π(710.29cm^3) = 961.12π cm^3

Finally, we will subtract the volume of the inside of the pot from the volume of the outside of the pot to find the volume of the clay:

V_clay = V_outer - V_inner = 961.12π cm^3 - 707.68π cm^3 = 253.44π cm^3

The capacity of the pot is 253.44π cm^3.

Note: The value of pi is 3.14 or approximately 3.14, you can use this value to calculate the capacity of the pot.

The expression that correctly calculates the perimeter of the shape is given as follows:

P = 2(6s + πr).

In which:

The perimeter of a square of side length s is given as follows:

P = 4s.

Hence, for three squares, the perimeter is given as follows:

P = 3 x 4s

P = 12s.

The perimeter, which is the circumference of a semicircle of radius r, is given by the equation presented as follows:

C = πr.

Hence the perimeter of two semicircles is given as follows:

C = 2πr.

The perimeter of the entire shape is given by the sum of the perimeter of each shape, hence:

P = 12s + 2πr.

P = 2(6s + πr).

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Answer: Part a= The probability of rolling a six on each roll is 1 in 6

So, a 100 in 600 chance but Julie has twice the probability.

So the die is most likely unfair,

Part b= Diagram is not attached to the question so to solve the first branch the probability of drawing show  each of the three colors from the five jellies

On the second draw, there are only four beans left and one of the three suits will be reduced by one. So while the probabilities of the first draw were fractions of 5, the probabilities of the second draw will be fractions of 4. The eight outcomes are the probability space of the two draws, which shows the composition of each event in the space and how likely that event is determined to be. Note that the sum of all eight probabilities is 1. You should be able to use this tree to find the answer to all the questions.

To know more about probability,

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