What is the result when the number 94 is decreased by 50%?

The given statement "Using p-value rule for a population proportion or mean, if the level of significance is less than p-value, null hypothesis is rejected." is True because the hypothesis is rejected in this case.

In hypothesis testing, the p-value is the probability of observing a test statistic as extreme as, or more extreme than, the observed test statistic, assuming that the null hypothesis is true. The level of significance, denoted by alpha, is the maximum probability of rejecting the null hypothesis when it is actually true.

If the p-value is less than the level of significance, it means that the observed test statistic is unlikely to have occurred by chance alone, assuming the null hypothesis is true. Therefore, we reject the null hypothesis in favor of the alternative hypothesis at the given level of significance.

For example, suppose we are testing the hypothesis that the population mean is equal to a certain value. If the p-value is 0.02 and the level of significance is 0.05, we would reject the null hypothesis because the p-value is less than the level of significance.

This means that there is strong evidence against the null hypothesis and we can conclude that the population mean is likely different from the hypothesized value.

In summary, if the level of significance is less than the p-value, we reject the null hypothesis in favor of the alternative hypothesis at the given level of significance.

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Answer:

2035.752 in³/min

Step-by-step explanation:

To find the rate of change of the volume of the sphere at the instant its radius is 9 inches, we need to work out dV/dt when r = 9.

The equation for the volume of a sphere is:

[tex]V=\dfrac{4}{3}\pi r^3[/tex]

Differentiate the expression for volume with respect to r:

[tex]\begin{aligned}V&=\dfrac{4}{3}\pi r^3\\\\\implies\dfrac{\text{d}V}{\text{d}r}&=3 \cdot \dfrac{4}{3} \pi r^{3-1}\\\\\dfrac{\text{d}V}{\text{d}r}&=4 \pi r^2\end{aligned}[/tex]

We know that that the radius of a sphere is increasing at a constant rate of 2 inches per minute, so the rate of change of the radius of sphere is:

[tex]\dfrac{\text{d}r}{\text{d}t}=2[/tex]

To find an expression for dV/dt, use the chain rule:

[tex]\boxed{\dfrac{\text{d}V}{\text{d}t}=\dfrac{\text{d}V}{\text{d}r} \times \dfrac{\text{d}r}{\text{d}t}}[/tex]

Substitute the expressions for dV/dr and dr/dt to create an expression for dV/dt:

[tex]\begin{aligned}\implies \dfrac{\text{d}V}{\text{d}t}&=4 \pi r^2 \times 2\\&=8 \pi r^2\end{aligned}[/tex]

To find the value of dV/dt when r = 9, substitute r = 9 into the equation for dV/dt:

[tex]\begin{aligned}\dfrac{\text{d}V}{\text{d}t}\;\textsf{at}\;r=9\implies \dfrac{\text{d}V}{\text{d}t}&=8 \pi (9)^2\\&=8\pi (81)\\&=648\pi\\&=2035.752\; \sf in^3/min\;(3\;d.p.)\end{aligned}[/tex]

Therefore, the rate of change of the volume of the sphere at the instant when its radius is 9 inches is 2035.752 in³/min (rounded to three decimal places).

The solution to the operations of similar triangles are:

ΔABC is similar to ΔEDC

They are similar triangles by AAA Similarity rule

x = 24

Similar triangles are defined as triangles that have the same shape, but their sizes may vary. Thus, if two triangles are similar, then their corresponding angles are congruent and corresponding sides are in equal proportion.

From the given image, we can see that:

AB is parallel to DE

Thus:

∠BAE ≅ ∠DCA because they are both alternate angles

Similarly:

∠ABD ≅ ∠EDB because they are both alternate angles

We can also say that:

∠ACB ≅ ∠DCE because they are both opposite angles

Thus: ΔABC is similar to ΔEDC

They are similar triangles by AAA Similarity rule

Using similar triangles ratio, we can say that:

x/15 =  16/10

x = (16 * 15)/10

x = 24

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The proportion of green lights is 0.4286.

(a) Yes, this situation is a Markov chain since the probability of transitioning to a new traffic light state only depends on the current state.

(b) Transition probability matrix:

G Y R

[tex]G 0.60 0.00 0.40[/tex]

[tex]Y 1.00 0.00 0.00[/tex]

[tex]R 0.30 0.00 0.50[/tex]

(c) There is only one class, as all states can be reached from any other state.

(d) After three more lights, the probability of being in each state is:

G Y R

G 0.3240 0.0000 0.6760

Y 0.5000 0.0000 0.5000

R 0.3150 0.0000 0.6850

(e) The proportion of green lights can be found by solving the steady-state equation. Assuming there are n traffic lights, the system of equations would be:

[tex]0.60g + 0.30r[/tex]= g

[tex]0.00g + 0.00r[/tex] = r

[tex]0.40g + 0.50r[/tex] = r

g + y + r = 1

Solving this system gives g =[tex]0.4286[/tex], y = [tex]0.1429[/tex], and r = [tex]0.4286.[/tex]

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Answer:

Step-by-step explanation:

This can be solved using Pythagorean Theorem.

a² + b² = c²

8383²  +  b²   = 158158²

70274689 + b² = 25013952964

70274689 - 70274689 + b² = 25013952964 - 70274689

    b² = 24943678275

√b²  = √24943678275

b = 157935.7 cm

Answer:I think the answer should be 22 because you subtract 27 the largest number from the smallest number which is 5 which gives you 22

Step-by-step explanation:

1) 27-5=22

The probability that the chosen integer is a perfect square or a perfect cube but not both is therefore 13/100.

When choosing an integer from 1 to 100, there are 10 perfect squares (1² to 10²) and 4 perfect cubes (1³ to 4³). However, the number 1 is both a perfect square and a perfect cube. To avoid counting it twice, we use the formula for the union of two sets: |A∪B| = |A| + |B| - |A∩B|. In this case, |A∪B| represents the number of integers that are either a perfect square or a perfect cube. Plugging in the values, we get |A∪B| = 10 + 4 - 1 = 13. The probability is therefore 13/100.

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The profit for the new container would be 3.47 each.

The gain from any business operation is referred to as profit. Every time a merchant sells a product, his goal is to make

a profit by getting something from the customer. In other words, if he sells the goods for more than the cost price, he

makes a profit; yet, if he needs to sell them for less, he loses money.

To calculate the profit for the new container, you need to follow these steps:

Determine the selling price. The marketing team suggests that the proposed container will sell for 3.97.

Determine the additional cost to make the new container. It will cost an additional 0.50 each to make.

Calculate the profit. Subtract the additional cost from the selling price to find the profit for the new container.

Profit = Selling Price - Additional Cost

Profit = 3.97 - 0.50

Profit = 3.47

So, the profit for the new container would be 3.47 each.

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Answer:

A)  The dimensions of the new triangular deck are:

B)  Area of original deck = 63 square feet
     Area of new deck = 393.75 square feet

C)  The ratio of the area is 4 : 25. This is the square of the scale factor.

Step-by-step explanation:

The given scale is 2 : 5. This means that the ratio of the measurements of the corresponding sides of two objects is 2 to 5. In other words, if one object has a length of 2 units, the corresponding length of the other object is 5 units.  So in this scenario, a 2 : 5 scaled version means that for every 2 foot of the original deck, there is 5 foot of the new deck.

Therefore, if the original base of the triangle is 18 feet, the new base, b, will be:

[tex]\implies \sf 2 : 5 = 18 : b[/tex]

[tex]\implies \sf \dfrac{2}{5} = \dfrac{18}{b}[/tex]

[tex]\implies \sf 2 \cdot b=18 \cdot 5[/tex]

[tex]\implies \sf b=\dfrac{18 \cdot 5}{2}[/tex]

[tex]\implies \sf b=45\;ft[/tex]

Similarly, if the original height of the triangle is 7 feet, the new height, h, will be:

[tex]\implies \sf 2 : 5 = 7 : h[/tex]

[tex]\implies \sf \dfrac{2}{5} = \dfrac{7}{h}[/tex]

[tex]\implies \sf 2 \cdot h=7 \cdot 5[/tex]

[tex]\implies \sf h=\dfrac{7\cdot 5}{2}[/tex]

[tex]\implies \sf h=17.5\;ft[/tex]

Therefore, the dimensions of the new triangular deck are:

[tex]\hrulefill[/tex]

The area of a triangle is half of the product of its base and height:

[tex]\boxed{\sf A=\dfrac{1}{2}bh}[/tex]

Area of the original deck:

[tex]\implies \sf A=\dfrac{1}{2} \cdot 18 \cdot 7[/tex]

[tex]\implies \sf A=9 \cdot 7[/tex]

[tex]\implies \sf A=63\;ft^2[/tex]

Area of the new deck:

[tex]\implies \sf A=\dfrac{1}{2} \cdot 45\cdot 17.5[/tex]

[tex]\implies \sf A=22.5 \cdot 17.5[/tex]

[tex]\implies \sf A=393.75\;ft^2[/tex]

Therefore, the areas of the two decks are:

[tex]\hrulefill[/tex]

The ratio of the area of the original deck to the area of the new deck is:

[tex]\implies \textsf{Area original deck}:\textsf{Area new deck}=\sf 63 : 393.75[/tex]

To rewrite the ratio in its simplest form, multiply both sides of the ratio by 16:

[tex]\implies \sf 63 \cdot 16: 393.75 \cdot 16=1008 :6300[/tex]

Then divide both sides of the ratio by 252:

[tex]\implies \sf \dfrac{1008}{252}:\dfrac{6300}{252}=4:25[/tex]

Therefore, the ratio of the area of the original deck to the area of the new deck in its simplest terms is 4 : 25.

If we compare this to the original scale factor 2 : 5 (which is the ratio of length), we can see that the ratio of area is the square of the scale factor:

[tex]\implies \sf 2^2:5^2=4:25[/tex]

The greatest common factor of 36x²y and 54xy²z is 18xy

The greatest common factor (GCF) of a set of numbers or polynomial is the largest factor that all the numbers/polynomial share.

Hence, let's find the greatest common factor of 36x²y and 54xy²z as follows:

Hence, let's find the greatest common factor of a set of polynomials which is the largest positive integer/variables that divides evenly into all numbers with zero as the remainder.

36x²y and 54xy²z

Therefore,

GCF = 18xy

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a. The probability that the amount of collagen is greater than 60 grams per mililiter is 0.5398

b. The probability that the amount of collagen is less than 90 grams per mililiter is 0.9985.

c. Approximately 68% of the data falls within 1 standard deviation of the mean for a normally distributed dataset.

a. To find the probability that the amount of collagen is greater than 60 grams per milliliter, we'll use the z-score formula:
z = (X - μ) / σ
where X is the value we're interested in, μ is the mean, and σ is the standard deviation.
z = (60 - 61) / 9.8 ≈ -0.1
Now, we'll use a z-table or calculator to find the probability corresponding to this z-score.

A z-score of -0.1 corresponds to a probability of approximately 0.4602.

Since we're looking for the probability that the amount of collagen is greater than 60 grams per milliliter, we need to find the area to the right of this z-score:
P(X > 60) = 1 - P(X ≤ 60) = 1 - 0.4602 ≈ 0.5398
b. To find the probability that the amount of collagen is less than 90 grams per milliliter, we'll again use the z-score formula:
z = (90 - 61) / 9.8 ≈ 2.96
A z-score of 2.96 corresponds to a probability of approximately 0.9985.

Since we're looking for the probability that the amount of collagen is less than 90 grams per milliliter, we'll use this probability directly:
P(X < 90) ≈ 0.9985
c. To find the percentage of compounds formed from the extract of this plant that fall within 1 standard deviation of the mean, we'll use the empirical rule (68-95-99.7 rule), which states that approximately 68% of the data falls within 1 standard deviation of the mean for a normally distributed dataset.

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According to the given question quotient of a given equation [tex](x^{3} +x^{2} +x+2)/(x^{2} -1)[/tex] is [tex]x[/tex] [tex]+ (3x+2)/ (x^{2} -1)[/tex]

When the roots and solutions of two equations coincide, the two equations are compared.

The same quantity, symbol, or expression must be added to or subtracted from both of the equation's two sides in order to produce an equivalent equation.

By multiplying or dividing each side of such an equation by a nonzero number, we can also produce a comparable equation.

Given

[tex](x^{3} +x^{2} +x+2)/(x^{2} -1)[/tex] by using polynomial long devision

After dividing we get

[tex]x+ (3x+2)/ (x^{2} -1)[/tex]

Therefore the given question quotient of a given equation [tex](x^{3} +x^{2} +x+2)/(x^{2} -1)[/tex] is [tex]x[/tex] [tex]+ (3x+2)/ (x^{2} -1)[/tex]

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After factorization we get:

14. (m²+ 4)(m + 2)(m - 2).

15. (x + 2y + 2z)(x + 2y - 2z).

16. m²(m + 1)(m - 1) - 12.

17. (2x + 3)² - 3(2x + 3).

Factorization is the process of breaking down a mathematical expression, equation, or number into its constituent parts or factors. In algebra, factorization involves finding the factors of a polynomial expression, which are the individual terms that can be multiplied together to yield the original polynomial.

14. To factor m⁴ - 16, we can use the difference of squares formula, which states that a² - b² = (a + b)(a - b). In this case, we can rewrite the expression as (m²)² - 4², and use the formula:

m⁴ - 16 = (m² + 4)(m² - 4)

Now, we can use the difference of squares formula again to factor m^2 - 4:

m⁴ - 16 = (m²+ 4)(m + 2)(m - 2)

15. x² + 4xy + 4y² - 4z² = (x + 2y)² - 4z²

(x + 2y)² - 4z² = (x + 2y + 2z)(x + 2y - 2z)

Therefore, the factored form of x² + 4xy + 4y² - 4z² is (x + 2y + 2z)(x + 2y - 2z).

16. To factor m⁴ + m² - 12, we can start by factoring out m^2:

m⁴ + m² - 12 = m²(m² + 1) - 12

Now we have a difference of squares in the parentheses, which we can factor using the formula a² + b² = (a + b)(a - b):

m²(m² + 1) - 12 = m²(m + 1)(m - 1) - 12

Therefore, the factored form of m⁴ + m² - 12 is m²(m + 1)(m - 1) - 12.

17. To factor (2x + 3)² + 2(2x + 3) - 15, we can recognize that it is a quadratic expression in the form of ax² + bx + c. We can first simplify the expression by using the distributive property:

(2x + 3)² + 2(2x + 3) - 15 = (2x + 3)(2x + 3) + 4x + 6 - 15

= (2x + 3)² + 4x - 9

Now, we can factor out the common factor of (2x + 3)² :

(2x + 3)² + 4x - 9 = (2x + 3)²+ 4x - 12x - 9

= (2x + 3)² - 8x - 9

= (2x + 3)² - (3)(3 + 2x)

Finally, we can factor out the common factor of (3 + 2x) from the two terms:

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

= (2x + 3)² - 3(2x + 3).

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The probability of selecting a blank marble on the first draw and an orange marble on the second draw is:

[tex](bo)/[(b+o)(b+o-1)][/tex]

Let B be the event that the first marble selected is blank and let O be the event that the second marble selected is orange. Since we are not replacing the marbles after each selection, the probability of the second marble being orange depends on the outcome of the first selection.

Let's assume that the bag contains b blank marbles and o orange marbles. The probability of selecting a blank marble on the first draw is b/(b+o), and the probability of selecting an orange marble on the second draw, given that the first marble was blank, is o/(b+o-1) (since there is one less marble in the bag after the first draw).

Therefore, the probability of selecting a blank marble on the first draw and an orange marble on the second draw is:

[tex]P(B\ and\ O )= P(B)*P(O|B)\\\\=b/(b+o)*o/(b+o-1)\\\\=(bo)/[(b+o)(b+o-1)][/tex]

Note that this assumes that there are no marbles that are both blank and orange. If there are such marbles in the bag, the calculation will be different.

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The correct area models are 4e and none of the given models for 4d and 4c.

What is the quadratic equation?

The solutions to the quadratic equation are the values of the unknown variable x, which satisfy the equation. These solutions are called roots or zeros of quadratic equations. The roots of any polynomial are the solutions for the given equation.

The area model of 4c is incorrect because the last term of the polynomial, which is 40, is represented by the bottom right box of the area model. However, in the given area model, the box representing 40 is in the upper row.

The area model of 4e is correct because it represents the polynomial as the product of two binomials, where the factors are (x-5) and (x-8).

When multiplied using the distributive property, the resulting polynomial is x² - 13x + 40, which matches the given polynomial form.

The area model of 4d is incorrect because it does not properly represent a factorable polynomial.

It only includes three boxes, representing the three terms of the polynomial, but it does not show how the terms can be factored into binomials.

Therefore, the correct area models are 4e and none of the given models for 4d and 4c.

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With D as Center of dilation, the scale factor is √2.

In mathematics, a scaling factor is a ratio that describes how much a figure or object has been enlarged or reduced in size. A scaling factor is typically expressed as a fraction or decimal that represents the relative change in size between the original figure and the new, scaled figure.

Now,

To find the scale factor of dilation, we can use the distance formula to find the ratio of corresponding sides in the original and dilated triangles.

The distance between points A and B in the original triangle is:

[tex]\rm AB = \sqrt{(x_2 - x_1)\² + (y_2 - y_1)\²}[/tex]

[tex]= \sqrt{(0 - (-1))\² + (1 - (-1))\²} \\\\= \sqrt{1\² + 2\²}[/tex]

= √5

The distance between points A' and B' in the dilated triangle is:

[tex]\rm A'B' = \sqrt{(x_2' - x_1')\² + (y_2' - y_1')\²}[/tex]

[tex]= \sqrt{(-2 - (-1))\² + (-5 - (-2))\²}[/tex]

[tex]= \sqrt{1\² + 3\²}[/tex]

= √10

Therefore, the scale factor for the dilation is:

Scale factor = A'B' / AB = √10 / √5 = √2

Hence, the scale factor of dilation is √2.

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By answering the presented question, we may conclude that equation Asymptotes: y = ±(3/9)x and y = ±(-3/9)x, which simplify to y = ±(1/3)x and y = ∓(1/3)x. and The hyperbola opens Vertically.

In mathematics, an equation is a statement that states the equivalence of two expressions. An equation is made up of two sides separated by an algebraic equation (=). For example, the argument "2x + 3 = 9" contends that the phrase "2x + 3" equals the number "9." The purpose of equation solving is to identify the value or values of the variable(s) that will make the equation true. Simple or complicated equations, regular or nonlinear, with one or more components are all possible. For example, in the equation[tex]"x2 + 2x - 3 = 0,[/tex]" the variable x is raised to the second power. Lines are utilized in many areas of mathematics, including algebra, calculus, and geometry.

Center: (0,0)

Transverse axis length: 2a = 2*3 = 6

Conjugate axis length: 2b = 2*9 = 18

Vertices: (0,±3)

Foci: (0,±√18)

Asymptotes: y = ±(3/9)x and y = ±(-3/9)x, which simplify to y = ±(1/3)x and y = ∓(1/3)x.

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Step-by-step explanation:

Acute angle means less than 90 degrees....so it is in the first Quadrant and sin and cos will be positive values

Trig identity

sin^2 + cos^2 = 1

sin^2 + .6^2 = 1    shows sin = .8

The tan  =   sin/ cos =  .8/.6 = 1.33

The product of equation (-a + 3)(a + 4) is -a² - a + 12.

What is product of equation?

Using the distributive property, we can expand the product of (-a + 3)(a + 4) as follows:

(-a + 3)(a + 4) = -a(a) - a(4) + 3(a) + 3(4)

Simplifying, we get:

(-a + 3)(a + 4) = -a² - 4a + 3a + 12

Combining like terms, we get:

(-a + 3)(a + 4) = -a² - a + 12

Therefore, the product of (-a + 3)(a + 4) is -a² - a + 12.

What is distributive property?

The distributive property is a property of arithmetic operations that is used to simplify expressions by distributing one term over another term inside parentheses.

In particular, the distributive property states that:

a × (b + c) = (a × b) + (a × c)

or

(a + b) × c = (a × c) + (b × c)

This property applies to both multiplication and addition, and can be extended to subtraction and division by adding or subtracting the distributive term to both sides of the equation.

For example, using the distributive property, we can simplify the expression:

3 × (4 + 5)

as follows:

3 × (4 + 5) = 3 × 4 + 3 × 5

= 12 + 15

= 27

In this example, we used the distributive property to distribute the term 3 over the sum (4 + 5), resulting in the equivalent expression 3 × 4 + 3 × 5.

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Answer:

The roots of p(z) are z = i, 2.91367 + 0.600499i, 1.22097 - 0.362212i, 0.306704 + 1.53771i.

Dr. Yanira's main issue is that basing her request on a single observation of a private school with the highest per-capita income may not be representative of the broader population of schools in the county, and the observation could be biased and not provide a true representation of the overall situation.

The main issue with Dr. Yanira basing her request for budget, school supplies, and school lunches, among other things, on a single observation of a private school with the highest per-capita income in the area is that the results may not be representative of the broader population of schools in the county.

The sample of one school may not be statistically significant or diverse enough to draw valid conclusions about the entire population of schools. The observation could be biased and may not provide a true representation of the overall situation. Therefore, it is important to use a larger and more diverse sample to draw more reliable conclusions.

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Answer:

The answer is $45

Step-by-step explanation:

$60 is dicounted at 25% off

25% percent of 60 is 15

60 - 15 = 45

A function which could be used to determine the total amount of revenue the company earns from selling moving boxes include the following: B. r(x) = -49x² + 567x.

From the information provided, the amount of revenue with respect to price that's being generated in this scenario can be calculated by using the following function (equation):

R(x) = x × P(x)

Where:

Since it is a revenue function, we would simply substitute the value of the unit price and evaluate as follows:

Revenue, R(x) = x × P(x)

P(0) = 567

By substituting the given parameters into the formula, we have;

Revenue, R(x) = x × (567 - 49x)

Revenue, R(x) = -49x² + 567x

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x=13, no idea what y is.

Answer:

Step-by-step explanation:

The formula to calculate the volume of a cone is:

v = h * π * r ^ 2/3

Where,

h: height

r: radio

We then clear the value of h:

h = (3 * v) / (π * r ^ 2)

We substitute the values:

h = (3 * (118π)) / (π * ((2/3) / (2)) ^ 2)

h = 3186 ft

Answer:

The height of the cone is:

h = 3186 ft

The solution to the system of equations is x = -1, y = 2.

A linear equation is a first-order (linear) term and a constant in an algebraic equation of the type y=mx+b, where m is the slope and b is the y-intercept. The previous equation, which has the variables y and x, is sometimes referred to as a "linear equation of two variables."

The adjective "linear" comes from the fact that the collection of solutions to such an equation forms a straight line in the plane.

These are the three types of linear equations:

By multiplying each equation by an appropriate constant, we may use the method of elimination to make the coefficients of x in both equations equal in size but opposite in sign. The following results from multiplying the first equation by 3 and the second equation by -2 in this situation:

a. 6x + 12y = 18

b. -6x - 10y = -14

To get rid of x, we can now combine these two equations:

a. 6x + 12y = 18

b. (-6x - 10y = -14)

2y = 4

By finding y, we obtain:

y = 2

We can find x by adding this value of y back into either of the initial equations:

2x + 4y = 6

2x + 4(2) = 6

2x + 8 = 6

2x = -2 x ⇒ -1

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the solution to the system of equations 2x+4y=6 and 3x+5y=7, after eliminating the x term, is x = -1 and y = 2.

The x term, we can multiply the first equation by -3 and the second equation by 2, so that the x term will have opposite coefficients and will cancel out when we add the two equations together.

[tex]-3(2x+4y=6) gives -6x - 12y = -18[/tex]

[tex]2(3x+5y=7) gives 6x + 10y = 14[/tex]

Adding these two equations gives:

[tex]-6x - 12y = -18[/tex]

[tex]+6x + 10y = 14[/tex]

[tex]-2y = -4[/tex]

Solving for y, we get:

[tex]y = 2[/tex]

Substituting this value of y back into either of the original equations, we can solve for x:

[tex]2x + 4y = 6[/tex]

[tex]2x + 8 = 6[/tex]

[tex]2x = -2[/tex]

[tex]x = -1[/tex]

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Hence, a cylinder with height h and base area [tex]\pi r^{2}[/tex] may be proven to have the same volume as a triangular prism.

Two parallel circular bases joined by a curving surface form the three-dimensional geometric object known as a cylinder. Due of its parallel and congruent bases, it is a sort of prism.

When someone uses the word "cylinder,” they often mean a right circular cylinder with circles for bases and an axis that is perpendicular to the bases' planes.

A cylinder's volume may be calculated using the formula V = [tex]r^{2}[/tex]h, where r denotes the perimeter of the base and h the height of the cylinder. L = 2rh, where r is the radius of a base and h is the height of the cylinder, is the formula for a cylinder's lateral surface area.

Cavalieri's principle states that if two solid objects have the same height and every cross-section taken perpendicular to a common axis has the same area, then the two objects have the same volume.

Let's consider the given triangular prism with base area  [tex]\pi[/tex][tex]r^2[/tex] and height h. If we take a cross-section perpendicular to the base at a height y, we get a circle with radius r multiplied by a triangle with base 2r and height y. The area of this cross-section is:

[tex]A(y) = \pi r^2 + (2r)(y) / 2[/tex]

Simplifying, we get:

[tex]A(y) = \pi r^2 + ry[/tex]

Now, let's consider a cylinder with base area pi r² and height h. If we take a cross-section perpendicular to the height at a height y, we get a circle with radius r multiplied by a rectangle with base 2r and height h. The area of this cross-section is:

[tex]A'(y) = \pi r^ + (2r)(h) = \pi r^2 + 2rh[/tex]

By Cavalieri's principle, the triangular prism and the cylinder have the same volume if A(y) = A'(y) for all y from 0 to h. Let's check:

[tex]\pi r^² + ry = \pi r^² + 2rh[/tex]

[tex]ry = 2rh[/tex]

[tex]y = 2r[/tex]

So, we see that the areas are equal for all y between 0 and h, except at [tex]y = 2r.[/tex] However, this is just a single point and has no effect on the volume,

Therefore,  the cylinder with base area [tex]\pi[/tex][tex]r²[/tex] and height h has the same volume as the triangular prism, with base area [tex]\pi[/tex]r² and height h.

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