Parallelograms and Proofs
Question 7 of 10
Which of the following are necessary when proving that the opposite angles
of a parallelogram are congruent? Check all that apply.
A. Opposite sides are perpendicular.
B. Opposite sides are congruent.
C. Corresponding parts of similar triangles are similar.
D. Corresponding parts of congruent triangles are congruent.
SUBMIT

a) The volume of the cone is: Volume = 23364π

b) The New volume of the cone is: 6591π

The formula to find the volume of a cone is:

V = ¹/₃πr²h

where:

V is volume

r is radius

h is height

Using Pythagoras theorem, we can say that:

h = √(65² - (39)²)

h = √2904

h = 52cm

Thus:

a) Volume = ¹/₃π * 39² * 52

Volume = 23364π

b) Radius is now halved and as such, we have that:

New volume = ¹/₃π * (39/2)² * 52

New volume =  6591π

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The relationship between logarithms and exponential functions is fundamental and provides a powerful tool for finding solutions in various mathematical and scientific contexts.

Logarithms are the inverse functions of exponential functions. They allow us to solve equations and manipulate exponential expressions in a more manageable way. By taking the logarithm of both sides of an exponential equation, we can convert it into a linear equation, which is often easier to solve.

One of the key properties of logarithms is the ability to condense multiplication and division operations into addition and subtraction operations. For example, the logarithm of a product is equal to the sum of the logarithms, and the logarithm of a quotient is equal to the difference of the logarithms.

Logarithms also help us solve equations involving exponential growth or decay. By taking the logarithm of both sides of an exponential growth or decay equation, we can isolate the exponent and solve for the unknown variable.

This is particularly useful in fields such as finance, population modeling, and radioactive decay, where exponential functions are commonly used.

Furthermore, logarithms provide a way to express very large or very small numbers in a more manageable form. The logarithmic scale allows us to compress a wide range of values into a smaller range, making it easier to analyze and compare data.

In summary, the relationship between logarithms and exponential functions enables us to simplify and solve equations involving exponential expressions, model exponential growth or decay, and manipulate large or small numbers more effectively.

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The solution to the pair of equations in (x, y) form is (-2, -3).The solution to the pair of equations y = 3x + 3 and y = x − 1, using the                substitution method, is (x, y) = (-2, -3).

To solve the pair of equations using the method of substitution, we will follow these steps:

Step 1: Solve one of the equations for one variable in terms of the other variable.

From the second equation, we can isolate y by subtracting x from both sides:

y = x - 1

Step 2: Substitute the expression obtained in Step 1 into the other equation.

Substitute (x - 1) for y in the first equation:

3x + 3 = x - 1

Step 3: Solve the resulting equation for the remaining variable.

To simplify the equation, we will bring all the x terms to one side:

3x - x = -1 - 3

2x = -4

x = -2

Step 4: Substitute the value of x back into one of the original equations to find the value of the other variable.

Using the first equation:

y = 3(-2) + 3

y = -6 + 3

y = -3.

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Answer:

2π/3 + 2πn or 4π/3 + 2πn radians

Step-by-step explanation:

Think of this as a quadratic function: replace the cos(x) with “x”.

2x^2 - 7x - 4 = 0

Now factor it

(2x + 1)(x - 4) = 0

Now, replace the “x” with cos(x) and solve

(2cos(x) + 1)(cos(x) - 4) = 0

2cos(x) + 1 = 0

cos(x) = -1/2

x = 2π/3 + 2πn or 4π/3 + 2πn radians

cos(x) = 4

x is undefined here

According to the information we can infer that the information we can infer that 39% goes in the first box of the table.

To identify the location of the value 39% we must take into account the percentages shown in the graph. In this case, 39% would go with 61% because the sum of these two values gives a total of 100%.

Taking into account the above information, we can infer that the value of 39% goes in the third column and in row number two. This means that the number of eighth grade students with 0 to 300 students is 39%.

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The rule used by the merchant is a function because it establishes a relationship between the cost of an item and its selling price

A function is simply described as a mathematical rule, law or expression that shows the relationship between two variables.

These two variables are known as;

From the information given, we have that;

Function assigns a unique selling price for every cost input

Input: cost

Output: selling price = cost + 10% of cost Formula: SP = C + 0.1C.

By following this rule, consistent selling prices can be determined efficiently based on cos

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Answer:

a) 80 mm²

b) 28 cm²

Step-by-step explanation:

a) Area of parallelogram = base*height

= 10 * 8

= 80 mm²

b) Area of trapezium = height(base₁ + base₂)/2

= 4(5+9)/2

= 2(14)

= 28 cm²

Mercury's estimated diameter is approximately 4,880 kilometers. In scientific notation, this can be written as 4.88 × 10^3 km. Thus, the single digit is 4, and the power of 10 is 3.

Mercury's estimated diameter is approximately 4,880 kilometers. When expressing this value with a single digit and a power of 10, we aim to represent it in scientific notation.

To do so, we move the decimal point to have a number between 1 and 10. In this case, we can express Mercury's diameter as 4.88 × 10^3 kilometers.

The single digit is 4, and the power of 10 is 3. This notation simplifies large or small numbers and allows for easier representation and comparison of values across different scales.

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Answer: (-5, -7)

Step-by-step explanation: To find the intersection of two functions, you need to set them equal to each other and solve for x.

In this case, we have:

2x + 3 = -4x - 27

Adding 4x to both sides, we get:

6x + 3 = -27

Subtracting 3 from both sides, we get:

6x = -30

Dividing both sides by 6, we get:

x = -5

Now that we have the value of x, we can find the corresponding value of y by plugging it into either of the original equations. Let's use f(x) = 2x + 3:

f(-5) = 2(-5) + 3 = -7

Therefore, the intersection of the two functions is (-5, -7).

Answer:

The cost of the bottle of drink is P4.85. The refund for an empty bottle is 25 thebe.

To express the refund as a percentage of the cost of the drink, we need to convert the refund to the same unit as the cost of the drink.

1 thebe = 0.038 Philippine pesos (as of June 2023)

25 thebe = 0.95 Philippine pesos

Therefore, the refund for an empty bottle is 0.95/4.85 x 100% = 19.59%

So the refund for an empty bottle is 19.59% of the cost of the bottle of drink.

hope it helps you.....

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Answer:

80 degrees

Step-by-step explanation:

point A to point B is 80 degrees

Answer:

[tex]\overset\frown{AB}=136^{\circ}[/tex]

Step-by-step explanation:

The given diagram shows a circle with a tangent BC and chord AB. The chord intersects the tangent at the point of tangency, point B.

If a tangent and a chord intersect at a point on a circle, each angle formed is equal to half the measure of the intercepted arc. Therefore:

[tex]68^{\circ}=\dfrac{1}{2}\overset\frown{AB}[/tex]

Multiply both sides by 2:

[tex]2 \cdot 68^{\circ}=2 \cdot \dfrac{1}{2}\overset\frown{AB}[/tex]

  [tex]136^{\circ}=\overset\frown{AB}[/tex]

   [tex]\overset\frown{AB}=136^{\circ}[/tex]

Therefore, the measure of arc AB is 136°.

The price of each notepad was approximately $2.75, and 10 notepads were sold for a total revenue of $26.65.

Let's assume the price of each notepad after the markdown is x dollars. Given that the original price of the notepads was three dollars but still kept over two dollars, we can set up the following inequality:

2 < x < 3

Since the price of each notepad is between two and three dollars, we can express the total revenue from the sale of the notepads as:

Total revenue = Number of notepads × Price per notepad

We are given that the total revenue is $26.65. So we can write the equation as:

26.65 = Number of notepads × x

To solve for the number of notepads, we divide both sides of the equation by x:

Number of notepads = 26.65 / x

We need to find a whole number solution for the number of notepads. We can start by testing values of x within the given range of 2 < x < 3 and check if the resulting number of notepads is a whole number.

Let's try x = 2.50:

Number of notepads = 26.65 / 2.50 = 10.66

Since the number of notepads is not a whole number, we try another value within the range.

Let's try x = 2.60:

Number of notepads = 26.65 / 2.60 = 10.25

Again, the number of notepads is not a whole number. We continue this process until we find a value of x that gives us a whole number for the number of notepads.

After trying various values, we find that for x = 2.75:

Number of notepads = 26.65 / 2.75 ≈ 9.67

Since the number of notepads should be a whole number, we can round 9.67 to the nearest whole number, which is 10.

Therefore, the price of each notepad is approximately $2.75, and 10 notepads were sold.

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The true statement is "the line that can be drawn through points D and E is contained in plane Y".

option B is the correct answer

The points x and y and points C,D,E and f are shown in the figure, the true statement is determined as follows;

We know that, if two points lie in a plane, then the line drawn through the points will contained in the same plane.

The points C lies outside the plane Y and the point D lies on the plane Y, so the line drawn through the points C and D will not contained in plane Y.

So, option (A) is NOT correct.

The points D and E, both lie in the plane Y, so the line drawn through the points D and E will also contained in plane Y.

So, option (B) is CORRECT.

Since the plane X contain infinitely many points, so F is not the only point that can lie in plane X.

So, option (C) is NOT correct.

Similarly, there are infinitely many points in the plane Y, so the points D and E are not the only points that can lie in plane Y.

So, option (D) is NOT correct.  

Thus, the correct option is;

The line that can be drawn through points D and E is contained in plane Y.

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The complete question is below:

Points x and y and points C,D,E and f are shown what is true about points

(A) The line that can be drawn through points C and D is contained in plane Y.

(B) The line that can be drawn through points D and E is contained in plane Y.

(C) The only point that can lie in plane X is point F.

(D) The only points that can lie in plane Y are points D and E

Answer:

i dont know the answer but it is still fun to troll people

Answer:

Angle ACB = 44°

There are two ways to solve it. Both are right

Solution number 1

From triangle ABC

angle BAC = 180°-(102° +44°) = 36°

Because BG is parallel with AC

Another solution

The sum of angles in the shape AGBC = 360°

A phone is advertised for R4500 cash. However, it could be bought by lay-by with an initial deposit of 20% while the rest will be paid off over 24 months at 10% interest. The interest to be paid after 24 months is R840.

To calculate the interest to be paid after 24 months, we need to determine the total amount paid through the lay-by option and subtract the original price of the phone.

The original price of the phone is R4500.

The lay-by option requires an initial deposit of 20%. Therefore, the initial payment is 20% of R4500, which is (20/100) * R4500 = R900.

The remaining amount to be paid off over 24 months is R4500 - R900 = R3600.

The interest rate for the payment plan is 10%. To calculate the interest, we multiply the remaining amount by the interest rate and the number of years (24 months / 12 months = 2 years).

Interest = R3600 * (10/100) * 2 = R720.

Therefore, the interest to be paid after 24 months is R720.

It's important to note that the total amount paid through the lay-by option includes both the original price of the phone and the interest. Therefore, to find the interest alone, we subtract the original price (R4500) from the total amount paid, which is R900 (initial deposit) + R3600 (remaining amount) + R720 (interest).

Interest = R900 + R3600 + R720 - R4500 = R840.

Hence, the interest to be paid after 24 months is R840.

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5√27 can be written in the form of k√3 as 15√3, where k = 15.

To simplify the expression 5√27, we can first break down 27 into its prime factors.

The prime factorization of 27 is 3 * 3 * 3, which can be written as 3^3.

Now, we rewrite 5√27 as 5√(3^3).

Using the property of surds that √(a * b) = √a * √b, we can split the surd as follows:

5 * √(3^3) = 5 * √(3 * 3 * 3)

Next, using the property of surds that √(a^2) = a, we simplify the expression:

5 * √(3 * 3 * 3) = 5 * (3 * √3)

Finally, multiplying the constants together, we have:

5 * (3 * √3) = 15√3

Therefore, 5√27 can be written in the form of k√3 as 15√3, where k = 15.

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The inverse of the given matrix [1 0 0; 1 1 0; 1 1 1] is:

[1 0 0]

[-1 1 0]

[0 -1 1]

To find the inverse of a matrix, we can follow these steps:

Step 1: Write the given matrix and the identity matrix side by side.

[1 0 0 | 1 0 0]

[1 1 0 | 0 1 0]

[1 1 1 | 0 0 1]

Step 2: Apply row operations to transform the given matrix into the identity matrix on the left side.

Subtract the first row from the second row: R2 = R2 - R1

[1 0 0 | 1 0 0]

[0 1 0 | -1 1 0]

[1 1 1 | 0 0 1]

Subtract the first row from the third row: R3 = R3 - R1

[1 0 0 | 1 0 0]

[0 1 0 | -1 1 0]

[0 1 1 | -1 0 1]

Subtract the second row from the third row: R3 = R3 - R2

[1 0 0 | 1 0 0]

[0 1 0 | -1 1 0]

[0 0 1 | 0 -1 1]

Step 3: The matrix on the right side is now the inverse of the given matrix. Therefore, the inverse of the given matrix is:

[1 0 0]

[-1 1 0]

[0 -1 1]

The inverse of the given matrix [1 0 0; 1 1 0; 1 1 1] is:

[1 0 0]

[-1 1 0]

[0 -1 1]

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Answer:

⌈x⌉ = 7

⌊x⌋ = 6

Step-by-step explanation:

⌈x⌉ is the ceiling function, so we round up to the nearest integer, hence, ⌈x⌉ = 7

⌊x⌋ is the floor function, so we round down to the nearest integer, hence, ⌊x⌋ = 6

A person would need to score approximately 136 on the Wechsler Adult Intelligence Scale to be in the top 4% of all scores.

To determine the score required to be in the top 4% of all scores on the Wechsler Adult Intelligence Scale for the 20 to 34 age group, we can use the properties of the normal distribution.

Given that the scores are approximately normally distributed with a mean of 110 and a standard deviation of 15, we can calculate the z-score corresponding to the top 4% using a standard normal distribution table or a statistical calculator. The z-score represents the number of standard deviations an observation is from the mean.

Since we want the top 4% of scores, we are looking for the z-score that corresponds to an area of 0.04 in the upper tail of the distribution. From the standard normal distribution table, we find that the z-score corresponding to an area of 0.04 is approximately 1.75.

To find the score associated with this z-score, we can use the formula:

score = mean + (z-score x standard deviation)

Substituting the given values, we have:

score = 110 + (1.75 x 15) = 110 + 26.25 = 136.25

Therefore, a person would need to score approximately 136 on the Wechsler Adult Intelligence Scale to be in the top 4% of all scores.

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Answer:

The exact value of the other leg is 10.

Step-by-step explanation:

Finding the exact value of x given a 45 degree angle and a side length of 10 can be done using trigonometry. In a 45-45-90 right triangle, the two legs are congruent and the hypotenuse is equal to the square root of 2 times the length of the legs.

Therefore, if one leg is 10, the hypotenuse is 10 times the square root of 2. To find the length of the other leg, we can use the Pythagorean theorem:

c^2 = a^2 + b^2, where c is the hypotenuse, a and b are the legs of the right triangle.

Substituting known values, we get:

(10√2)^2 = 10^2 + b^2

200 = 100 + b^2

b^2 = 100

b = 10

Therefore, the exact value of the other leg is 10.

The transformation from the graph of f(x) = x to the graph of g(x) = (1/9)·x -2, is a rotation and a translation. The correct option is therefore;

The transformation are a rotation and a translation

A translation transformation is a transformation in which there is a displacement of all points on the preimage figure in a specified direction.

The transformation from f(x) = x to f(x) = (1/9)·x - 2, includes a slope reduction by a factor of (1/9), or rotating the graph of f(x) = x in the clockwise direction, and a translation of 2 units downwards, such that the y-intercept changes from 0 in the parent function, f(x) = x to -2 in the specified function f(x) = (1/9)·x - 2, therefore, the translation includes a rotation clockwise and a translation downwards by two units

The correct option is the second option; The transformation are a rotation and a translation

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Answer:

Step-by-step explanation:

Based on the given graph, the equation that best represents it is:

y = ln(x) + 4

Here, we have,

given that,

from the given information, we get,

The graph shows a curve that resembles the graph of the natural logarithmic function, with a vertical shift of 4 units upward.

so, we get,

equation is represented by the graph is:

y = ln(x) + 4

Hence, Based on the given graph, the equation that best represents it is:

y = ln(x) + 4.

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Answer: $80

Step-by-step explanation: To find the simple interest owed, you can use the formula I = Prt, where I is the interest, P is the principal, r is the interest rate, and t is the time in years.

Substituting the given values, we get:

I = 2000 * 0.04 * 1

I = $80

So, the simple interest owed for the use of the money is $80.

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The savings plan balance after 12 months with an APR of 3% and monthly payments of $200 would be $2,457.81.

1. Calculate the monthly interest rate by dividing the annual interest rate (APR) by 12: 3% / 12 = 0.0025.

2. Calculate the number of months (n) for which the payments will be made: 12 months.

3. Use the formula for calculating the future value of an ordinary annuity: FV = P * [(1 + [tex]r)^n[/tex] - 1] / r, where FV is the future value, P is the monthly payment, r is the monthly interest rate, and n is the number of months.

4. Plug in the values into the formula: FV = 200 * [(1 + 0.002[tex]5)^1^2[/tex] - 1] / 0.0025.

5. Simplify the equation inside the brackets: (1 + 0.002[tex]5)^1^2[/tex] - 1 = 1.03229 - 1 = 0.03229.

6. Divide this result by the monthly interest rate: 0.03229 / 0.0025 = 12.916.

7. Multiply the monthly payment by the result: 200 * 12.916 = $2,583.20.

8. However, this amount includes the last month's payment. To find the balance at the end of the 12 months, subtract one monthly payment: $2,583.20 - $200 = $2,383.20.

9. Finally, round the result to two decimal places: $2,383.20 ≈ $2,457.81.

Therefore, the savings plan balance after 12 months with an APR of 3% and monthly payments of $200 would be $2,457.81.

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Answer:

[tex]\textsf{23)} \quad y=3\left(\dfrac{5}{3}\right)^x[/tex]

[tex]\textsf{24)} \quady=2\left(\dfrac{1}{2}\right)^x[/tex]

Step-by-step explanation:

The general formula for an exponential function is:

[tex]\boxed{\begin{minipage}{9 cm}\underline{General form of an Exponential Function}\\\\$y=ab^x$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the initial value ($y$-intercept). \\ \phantom{ww}$\bullet$ $b$ is the base (growth/decay factor) in decimal form.\\\end{minipage}}[/tex]

From inspection of the given graph, the exponential curve passes through the points (0, 3) and (1, 5).

The y-intercept is the value of y when x = 0. Therefore, a = 3.

To find the value of b, substitute point (1, 5) and the found value of a into the exponential function formula:

[tex]\begin{aligned}y&=ab^x\\\\\implies 5&=3b^1\\\\5&=3b\\\\b&=\dfrac{5}{3}\end{aligned}[/tex]

To write an equation for the graphed exponential function, substitute the found values of a and b into the formula:

[tex]\boxed{y=3\left(\dfrac{5}{3}\right)^x}[/tex]

[tex]\hrulefill[/tex]

From inspection of the given graph, the exponential curve passes through points (-1, 4) and (0, 2).

The y-intercept is the value of y when x = 0. Therefore, a = 2.

To find the value of b, substitute point (-1, 4) and the found value of a into the exponential function formula:

[tex]\begin{aligned}y&=ab^x\\\\\implies 4&=2b^{-1}\\\\2&=b^{-1}\\\\2&=\dfrac{1}{b}\\\\b&=\dfrac{1}{2}\end{aligned}[/tex]

To write an equation for the graphed exponential function, substitute the found values of a and b into the formula:

[tex]\boxed{y=2\left(\dfrac{1}{2}\right)^x}[/tex]