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Answer : 1 and 3 are the correct probabilities.

→According to the given Venn diagram.

Total number of elements  = 59.

1)P(C)=[tex]\frac{21}{59}[/tex] and [tex]P(A\cap C)=\frac{14}{59}[/tex] then

[tex]P(A|C)=\frac{P(A\cap C)}{P(C)}[/tex][tex]=\frac{\frac{14}{59}}{\frac{21}{59}}=\frac{14}{21}=\frac{2}{3}[/tex]

2)P(B)=[tex]\frac{27}{59}[/tex] and  [tex]P(C\cap B)=\frac{11}{59}[/tex] then

[tex]P(C|B)=\frac{P(C\cap B)}{P(B)}[/tex][tex]=\frac{\frac{11}{59}}{\frac{27}{59}}=\frac{11}{27}[/tex][tex]\neq \frac{8}{27}[/tex]

3) P(A) =[tex]\frac{number\ of\ elements\ in\ A}{Total\ elements}=\frac{31}{59}[/tex]

4) P(C) =[tex]\frac{number\ of\ elements\ in\ C}{Total\ elements}=\frac{21}{59}[/tex][tex]\neq \frac{3}{7}[/tex]

5) [tex]P(B|A)=\frac{P(B\cap A)}{P(A)}[/tex][tex]=\frac{\frac{13}{59}}{\frac{31}{59}}=\frac{13}{31}[/tex][tex]\neq \frac{13}{27}[/tex]

Therefore, option 1 and 3 are correct.

Answer:

11

Step-by-step explanation:

Cost function: [tex]\( nc = 72 \)[/tex]. Cost per person for 12 students: $6. Answer: C.

To determine the function that models the data and to find the cost per person if 12 students go on the trip, we need to analyze the relationship between the number of students (n) and the cost per student (c).

Given the data:

- When [tex]\( n = 3 \), \( c = 24 \)[/tex]

- When [tex]\( n = 6 \), \( c = 12 \)[/tex]

- When [tex]\( n = 9 \), \( c = 8 \)[/tex]

- When [tex]\( n = 16 \), \( c = 4.5 \)[/tex]

We can observe that as the number of students increases, the cost per student decreases. This suggests an inverse relationship between the number of students and the cost per student. The form of an inverse relationship can be expressed as:

[tex]\[ c = \frac{k}{n} \][/tex]

where [tex]\( k \)[/tex] is a constant.

To find the constant [tex]\( k \)[/tex], we can use one of the data points. Let's use the first data point ([tex]\( n = 3 \), \( c = 24 \)[/tex]):

[tex]\[ 24 = \frac{k}{3} \][/tex]

Solving for [tex]\( k \)[/tex]:

[tex]\[ k = 24 \times 3 = 72 \][/tex]

So the function that models the data is:

[tex]\[ c = \frac{72}{n} \][/tex]

Now, we need to find the cost per person if 12 students go on the trip. We substitute [tex]\( n = 12 \)[/tex] into the function:

[tex]\[ c = \frac{72}{12} = 6 \][/tex]

Therefore, the cost per person if 12 students go on the trip is $6.

The correct answer is:

C. [tex]\( nc = 72 \)[/tex], $6

To confirm this, we can check that this function fits all the provided data points:

1. For [tex]\( n = 3 \)[/tex]:

[tex]\[ c = \frac{72}{3} = 24 \][/tex] (matches the given cost)

2. For [tex]\( n = 6 \)[/tex]:

[tex]\[ c = \frac{72}{6} = 12 \][/tex] (matches the given cost)

3. For [tex]\( n = 9 \)[/tex]:

[tex]\[ c = \frac{72}{9} = 8 \][/tex] (matches the given cost)

4. For [tex]\( n = 16 \)[/tex]:

[tex]\[ c = \frac{72}{16} = 4.5 \][/tex] (matches the given cost)

Hence, the function [tex]\( c = \frac{72}{n} \)[/tex] is validated by all the data points.

Solution:

we are given with [tex] (a + b)(a^2 - ab + b^2) [/tex]

we have been asked to find the representation of the given factor as a Sum.

As we know that

[tex] (a^3+b^3)= (a + b)(a^2 - ab + b^2) [/tex]

Because

[tex] (a + b)(a^2 - ab + b^2)=a(a^2 - ab + b^2)+b(a^2 - ab + b^2)\\
\\
(a + b)(a^2 - ab + b^2)=a^3-a^2b+ab^2+ba^2-ab^2+b^3\\
\\
(a + b)(a^2 - ab + b^2)=a^3+b^3\\ [/tex]

The area of the larger quadrilateral is found by using the scale factor derived from the ratio of the perimeters of the similar quadrilaterals, which comes out to 150 cm².

The question involves finding the area of a larger quadrilateral given the perimeters of two similar quadrilaterals and the area of the smaller one. In similar figures, the ratio of their areas is the square of the ratio of their corresponding linear dimensions, such as side lengths or perimeters in this case. Since the perimeters are 48 cm and 60 cm, the ratio of the perimeters (also the scale factor) is 48/60, or 4/5. Consequently, the ratio of the areas is the square of the scale factor, which is (4/5)² or 16/25. Knowing the area of the smaller quadrilateral is 96 cm², we find the area of the larger quadrilateral by dividing the area of the smaller by the scale factor squared and then multiplying by the larger scale factor squared. This gives us (96 / (16/25)) cm² = (96 * 25/16) cm² = 150 cm².

So, the area of the larger quadrilateral is 150 cm².

Final answer:

By creating and solving a system of equations based on the given information, we find that there are 10 blue marbles and 8 red marbles.

Explanation:

To solve this problem, we set up two equations based on the information given:

3 times the number of blue marbles (3B) exceeds 2 times the number of red marbles (2R) by 18: 3B - 2R = 18.

5 times the number of blue marbles (5B) is 2 less than 6 times the number of red marbles (6R): 5B = 6R - 2.

Now, we can solve these equations using substitution or elimination. Using substitution, solve the second equation for B:

B = ⅜(6R - 2)

Then substitute this expression for B in the first equation:

3(⅜(6R - 2)) - 2R = 18

Solve for R:

R = 8

Now substitute R back into the equation for B:

B = ⅜(6(8) - 2) = 10

Thus, there are 10 blue marbles and 8 red marbles.

In isosceles trapezoids, the diagonals are not perpendicular. They are congruent, the bases are parallel, and the non-parallel sides are congruent.

An isosceles trapezoid is a type of quadrilateral that has a pair of parallel sides, known as the bases, and the other two sides, not parallel, are of equal length. The statement 'The diagonals are perpendicular' is NOT true for isosceles trapezoids. In isosceles trapezoids, the diagonals are congruent and not perpendicular. Just to put in context, the perpendicular diagonals are a characteristic of rhombuses and not of isosceles trapezoids.

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The answer should be 7092.822912 or 7092.82 when rounded to the nearest hundredth because the formula for volume is V= π times r^2 times height to get the answer. So: 3.14 x 9.8^2 x 23.52, and that's the answer.

QUESTION 1

We want to find the volume of a circular cylinder with a diameter of [tex]19.6yd[/tex] and a height of [tex]23.52yd[/tex].

The volume of a cylinder is given by the formula

[tex]V=\pi r^2h[/tex]

where [tex]h=23.52yd[/tex] and [tex]r=9.8yd[/tex] is half the diameter of the cylinder and [tex]\pi=3.14[/tex].

We substitute all these values into the formula to obtain,

[tex]V=3.14\times 9.8^2\times 23.52[/tex]

[tex]V=7092.82[/tex] square yards to the nearest hundredth.

QUESTION 2

We want to find the volume of a right circular cylinder with a base diameter of [tex]18yd[/tex] and a height of [tex]3yd[/tex].

The volume of a cylinder is given by the formula

[tex]V=\pi r^2h[/tex]

where [tex]h=3yd[/tex] and [tex]r=9yd[/tex] is half the diameter of the cylinder.

We substitute all these values into the formula to obtain,

[tex]V=\pi \times 9^2\times 3[/tex]

[tex]V=243\pi[/tex] square yards.

Final answer:

The ratio of 1.02 can be converted to a fraction by multiplying by 100 to get 102/100 and then simplifying to its simplest form, which is 51/50.

Explanation:

To write the ratio 1.02 as a fraction in its simplest form, we first recognize that 1.02 is the same as 1.02/1. If we want to express this as a fraction, we must remove the decimal point by multiplying both the numerator and the denominator by 100 (because there are two digits after the decimal point). This gives us 102/100. We can then simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2. Therefore, the simplified fraction is 51/50.

The hypotenuse of the isosceles right triangle is [tex]\( 12 \)[/tex] units.

In an isosceles right triangle, the legs are congruent, and the hypotenuse can be found using the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse [tex]\( c \)[/tex] is equal to the sum of the squares of the lengths of the other two sides [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:

[tex]\[ c^2 = a^2 + b^2 \][/tex]

Given that each leg of the isosceles right triangle has a measure of [tex]\( 6\sqrt{2} \)[/tex], we can substitute this value into the formula:

[tex]\[ c^2 = (6\sqrt{2})^2 + (6\sqrt{2})^2 \]\[ c^2 = 36 \times 2 + 36 \times 2 \]\[ c^2 = 72 + 72 \]\[ c^2 = 144 \][/tex]

Now, we take the square root of both sides to find the length of the hypotenuse [tex]\( c \):[/tex]

[tex]\[ c = \sqrt{144} \][/tex]

[tex]\[ c = 12 \][/tex]

So, the hypotenuse of the isosceles right triangle is [tex]\( 12 \)[/tex] units.

Answer:

Step-by-step explanation:

Amps refers to the electrical current.

We have to find the electrical current when the car has a 12-volt battery and the engine has a resistance of 0.22 ohms.

The relation between these magnitudes is

[tex]V=I\times R[/tex]

Where [tex]V[/tex] is the voltage, [tex]I[/tex] is the electrical current and [tex]R[/tex] the resistance.

We know that [tex]V=12; R=0.22[/tex]. Replacing these values in the formula and solving for [tex]I[/tex]

[tex]V=I\times R\\\frac{V}{R}=I\\ I=\frac{12}{0.22}\\ I\approx 54.54 amp[/tex]

Therefore, the answer is around 54.54 amps.

The current drawn from the battery when the key is turned is approximately 54.55 amps.

To determine the current drawn from the battery, we can use Ohm's Law, which states that the current (I) is equal to the voltage (V) divided by the resistance (R):

[tex]\[ I = \frac{V}{R} \][/tex]

Given that the voltage (V) of the battery is 12 volts and the resistance (R) of the engine is 0.22 ohms, we can plug these values into the equation:

[tex]\[ I = \frac{12 \text{ volts}}{0.22 \text{ ohms}} \][/tex]

[tex]\[ I = \frac{12}{0.22} \][/tex]

[tex]\[ I \approx 54.5454 \][/tex]

Rounding to the nearest hundredth, we get:

[tex]\[ I \approx 54.55 \text{ amps} \][/tex]

Expressions with exponents can be simplified using rules of exponents, and numbers can be converted into scientific notation by recognizing how to move the decimal point and denote magnitude with the power of ten.

To simplify expressions with exponents and convert numbers into scientific notation, we apply the rules of exponents and understand the format of scientific notation.

(2t)⁻⁶: Using the negative exponent rule, which states that a⁻⁶ = 1/a⁶, we can simplify this expression to 1/(2⁶t⁶).

(w⁻²j⁻⁴)⁻³(j⁷j³): To deal with the negative and compounded exponent, we invert and take the cube, resulting in w⁶j¹². Then, multiply the j terms together to get w⁶j¹⁵.

To simplify a²b⁻⁷c⁴ / a⁵b³c⁻², we subtract exponents when dividing like bases, resulting in a⁻³b⁻¹°c⁶.

For the expression z⁻ᵗ(mᵗ)¹, when any variables are raised to the zero power, the result is 1. Thus, the entire expression evaluates to 1 due to (mᵗ)¹ becoming 1.

Converting to scientific notation: To express 0.0002603 in scientific notation, it becomes 2.603 × 10⁻⁴. The number 5.38 × 10² in standard notation is 538.

By applying these step-by-step procedures, we can simplify expressions using positive exponents and accurately convert between standard notation and scientific notation.

The value of a in the diagram is 144°.

The correct option is B.

An open, flat object with four straight sides and one set of parallel sides is referred to as a trapezoid or trapezium. A trapezium's non-parallel sides are referred to as the legs, while its parallel sides are referred to as the bases.

Given:

A trapezium has parallel bases and that one of its characteristics is that the two angles on a single side are supplementary, meaning that the total of the angles on two neighboring sides is 180°.

So,

∠a = 180 - 36

∠a = 144

∠b = 180 - 113

∠b = 67

Therefore, the value of a is 144°.

To learn more about the trapezoid;

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The complete question is given in the attached image.

Answer:

540

Step-by-step explanation: