A block weighing 25№ has dimensions 34cm * 25 *
10em. what is the greatest pressure and the least.
the ground?
pressure it can
exert on

The area of the figure is approximately 86.325 square centimeters.

A triangle is a shape with three sides and three angles. It is one of the basic shapes in geometry. Triangles can be classified according to the lengths of their sides as equilateral, isosceles and scalene triangles, and also according to their internal angles as right, acute and obtuse triangles.

To find the area of the trapezoid triangle , we need to first find the length of the other diagonal

Since the center line is perpendicular to both diagonals, it divides the trapezoid triangle into two right triangles, ABC and ABD.

Using the Pythagorean theorem, we can find the length of BD:

BD = [tex]\sqrt{(AB^2 + AD^2)}[/tex]

BD = [tex]\sqrt{(10.2^2 + 7.5^2)}[/tex]

BD = 12.654

Now, we can use the formula for the area of a trapezoid:

A = (1/2) * (AB + CD) * h

where h is the height of the trapezoid, which is the distance between the parallel sides AB and CD.

To find h, we can use the formula for the area of a triangle:

A = (1/2) * b * h

where b is the base of the triangle and h is its height.

We can use triangle ABD, which is a right triangle, to find the height h:

h = AD = 7.5

Now we can substitute the values into the formula for the area of a trapezoid:

A = (1/2) * (AB + CD) * h

A = (1/2) * (10.2 + 15.3) * 7.5

A = 86.325

Therefore, the area of the trapezoid triangle ABC is approximately 86.325 square centimeters.

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Answer:

The correct answer is B) rhombus, rectangle, and square.

A parallelogram is defined as a quadrilateral with two pairs of parallel sides. A rhombus, a rectangle, and a square are all special cases of parallelograms with additional properties.

A rhombus is a parallelogram with all sides equal in length.

A rectangle is a parallelogram with four right angles.

A square is a parallelogram with four equal sides and four right angles.

A kite and a trapezoid are not parallelograms since they do not have two pairs of parallel sides. While a trapezoid has one pair of parallel sides, a kite has no parallel sides.

Twο statement for this graph are true -

O The functiοn is increasing οn the interval (-∞, 0).

O The functiοn is increasing οn the interval (0, ∞).

A functiοn's parameters are a grοup οf values that can be plugged intο the dοmain οf the functiοn. This set οf data cοntains the x values fοr a functiοn like f. (x).

The term "range" refers tο the set οf numbers that the functiοn cοnsiders tο be valid. The functiοn οutputs this string οf values in respοnse tο the insertiοn οf an x value.

The functiοn h(x) = 3x is a linear functiοn (straight line) with a slοpe οf 3, and a y-intercept οf 0.  The cοnstant, pοsitive slοpe means that h(x) will increase οver the interval (-∞, ∞).

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The complete question is -

Given the function h(x)=3x, which statement is true about h(x)?

O The function is decreasing on the interval (-∞, 0).

O The function is increasing on the interval (-∞, 0).

O The function is decreasing on the interval (0, ∞).

O The function is increasing on the interval (0, ∞).

Therefore, we need 7.4 ounces of rye flour and 29.6 ounces of whole-wheat flour to make a loaf of bread weighing 37 ounces, according to the recipe.

An equation is a mathematical statement that shows the equality of two expressions, usually with an equal sign "=" in between them. An equation can contain variables, constants, and operators. The variables are represented by letters and can take on different values, while constants are fixed values that do not change. Operators include mathematical symbols like plus, minus, multiplication, and division, as well as exponents and logarithms.

Here,

Let's first assign variables to represent the amounts of rye and whole-wheat flour we need to use:

Let x be the number of ounces of rye flour.

Let y be the number of ounces of whole-wheat flour.

We know from the recipe that we need to use 1 oz of rye flour for every 4 oz of whole-wheat flour. This can be expressed as:

x = y/4 (equation 1)

We also know that the total weight of the loaf of bread should be 37 oz. This means that:

x + y = 37 (equation 2)

Now we can use equation 1 to substitute for x in equation 2:

y/4 + y = 37

Multiplying both sides by 4 to eliminate the fraction, we get:

y + 4y = 148

5y = 148

y = 29.6

So we need 29.6 ounces of whole-wheat flour. Using equation 1 to find x:

x = y/4 = 29.6/4 = 7.4

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Therefore , the solution of the given problem of probability comes out to be rotating blue has two positive results.

Each of the procedure's criteria-based methods have as their main objective determining the likelihood that a statement is accurate or that an event will take place. Any number from zero to one, where 0 typically represents percentage and 1 typically reflects degree of certainty, can be used to symbolize chance. A probability illustration shows the likelihood that a particular occurrence will occur.

Here,

A. The number of potential outcomes for each spin can be multiplied to determine the number of outcomes in the sample space. Since there are two potential outcomes for each spin in this scenario, there are a total of:

=> 2 x 2 = 4

There are four potential outcomes in the sample space as a result.

B. There are two of these branches: blue-blue and red-red. Therefore, spinning the same hue twice in a row has two positive results.

C. Counting the number of branches that lead to blue on either spin will allow us to determine how many good outcomes there are for spinning blue. These stems come in two varieties: blue-red and blue-blue. Therefore, rotating blue has two positive results.

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The quadratic function on the graph is:

y = (-4/9)*(x - 3)^2 + 6

Here we can see that we have a quadratic equation, remember that if the vertex is (h, k) and the leading coefficient is a, then we can write this as:

f(x) = a*(x - h)^2 + k

Here we can see that the vertex is at (3, 6), then we can write:

f(x) = a*(x - 3)^2 + 6

And we can see that the y-intercept is (0, 2), then we can write:

2  =a*(0 - 3)^2 + 6

2 = a*9 + 6

2 - 6 = a*9

-4/9 = a

Then the function is:

y = (-4/9)*(x - 3)^2 + 6

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The stem and leaf plot for the data given is obtained as -

Stem  Leaf

0         3 3 3 5 6

1          2 2 3 3 3 4 5 9

2         2 2 2 3 3

3         7 7 9  

4         3

What is stem and leaf plot?

A stem and leaf plot, also known as a stem and leaf diagram, is a way to arrange data so that it is simple to see how frequently various sorts of values occur. It is a graph that displays ordered numerical data. A stem and a leaf are divided into each piece of data.

The data is given that represents the number of grams of fat in breakfast meals offered at a local fast food restaurant.

According to the data, the stem and leaf plot is -

Stem  Leaf

0         3 3 3 5 6

1          2 2 3 3 3 4 5 9

2         2 2 2 3 3

3         7 7 9  

4         3

The stem contains the first digits of the number, and the leaf plot contains the second digits of the number.

Therefore, the stem and leaf plot is obtained.

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Answer:

Bailey earned approximately $499.99 on Thursday, including his commission

Step-by-step explanation:

Bailey's earnings on Thursday consist of his base pay plus his commission. His base pay is $90, and his commission is 11% of his sales.

Let's call Bailey's sales on Thursday "S". Then his commission is 0.11S.

We know that Bailey's total earnings on Thursday were $500. So we can write an equation:

$90 + 0.11S = 500$

To solve for S, we can subtract $90$ from both sides of the equation:

$0.11S = 410$

Then we can divide both sides by $0.11$:

$S = 3,727.27$

So Bailey's sales on Thursday were $3,727.27.

To find Bailey's total earnings, we can add his base pay and commission:

$90 + 0.11(3,727.27) \approx 499.99$

So Bailey earned approximately $499.99 on Thursday, including his commission.

Comparing the three in their simplest form we have the smallest ratio as 3 to 4.

A mathematical phrase that compares two values is called a ratio. It is written as the product of the division of two different quantities. As in 2:1 or 2/1, ratios are frequently expressed as a fraction or with a colon.

Mathematicians employ ratios in many different contexts, such as arithmetic, algebra, and geometry. They are used to compare amounts, identify equivalent values, and resolve proportional and rate-of-change issues.

Convert the ratios in their simplest form:

3 to 4 can be simplified by dividing both terms by their greatest common factor, which is 1. Therefore, 3 to 4 is already in its simplest form.

10 to 12 can be simplified by dividing both terms by their greatest common factor, which is 2. So, 10 to 12 simplifies to 5 to 6.

2 to 1 is already in its simplest form.

Thus, comparing the three we have the smallest ratio as 3 to 4.

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762 base 10 to base 7 is 2136 base 7

Base Numbers are the number of units, or numbers, we use in our counting system. The most common base number is ten because the digits 0-9 are used in the number system.

And  it is a base-7 number system we would only use seven different digits, like 0-6.

How to determine this

By dividing the number 762 base 10 in base 7 until it can no more be divided in base 7.

First divide 762 /7 = 108 remainder 6

108/7 = 15 remainder 3

15/7 = 2 remainder 1

2/7 = 0 remainder 2

Where the remainders arranged in descending is the base system.

So, by arranging from the bottom

= 2136 base 7

Therefore, 762 base 10 is 2136 base 7

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The circumference of the circle is 150.768 units.

The area of the circle is 1809.216 units squared.

The area and circumference of a circle can be found as follows:

Let's find the circumference of the circle as follows:

circumference of a circle =  2πr

where

Therefore,

circumference of a circle = 2πr

r = 24 units

π = 3.141

Hence,

circumference of the circle = 2 × 3.141 × 24

circumference of the circle =  150.768 units

Let's find the area of the circle as follows:

area of the circle = πr²

area of the circle = 3.141 × 24²

area of the circle = 1809.216 units squared

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Answer:

18:24

Step-by-step explanation:

We know

For every 7 apples, we divided to 3:4

To get from 7 to 42, we time 6. So, we take the ratio times 6 and get 18:24

That the gradient gives the direction of steepest ascent, so the negative of the gradient gives the direction of steepest descent.

To find the slope of the tangent in the positive y-direction to the surface z = 3x + 2y − 4exy at (6, 0, 14), we need to find the partial derivative of z with respect to y and evaluate it at the point (6,0):

∂z/∂y = 2 - 4ex(6)

At (6,0), this becomes:

∂z/∂y = 2 - 4e(36) = -646.158

So the slope of the tangent in the positive y-direction at (6,0,14) is -646.158.

To find the following for z = x2 + 5x − 3y3:

a) The partial derivative of z with respect to x:

∂z/∂x = 2x + 5

b) The partial derivative of z with respect to y:

∂z/∂y = -9y2

c) The gradient of z:

∇z = (2x + 5)i - 9y2j

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After addressing the issue at hand, we can state that As a result, x equals  triangle 180 degrees.

A polygon is a triangle with four or more sections. It is rectangular in shape. The edges of a triangle ABC are A, B, and C. Euclidean geometry yields a single plane and cube when the sides are not collinear. A polygon is formed when a triangle has three components and three angles. The corners of a triangle are the points at which the three edges meet. Triangle sides add up to 180 degrees.A polygon is a triangle with four or more sections. It is rectangular in shape. The edges of a triangle ABC are A, B, and C. Euclidean geometry yields a single plane and cube when the sides are not collinear. A polygon is formed when a triangle has three components and three angles. The corners of a triangle are the points at which the three edges meet. Triangle sides add up to 180 degrees.

To calculate x, we must use the fact that the sum of a triangle's interior angles is 180 degrees.

First, determine the size of angle B. Angle A is 60 degrees, and angle C is 75 degrees. So,

angle B = 180 minus angle A A - angle C angle B = 180 - 60 - 75 angle B = 45

D = 360 minus angle angle A angle B angle C angle D = 360° - 60° - 45° - 75° D is equal to 180 degrees.

angle E stands for angle. D\sangle E is equal to 180 degrees.

As a result, the value of x is:

x = the angle 180 degrees = E x

As a result, x equals 180 degrees.

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Answer:

We can find the value of f(-2) by substituting -2 for x in the given function:

f(-2) = 2(-2)^(2) - 3(-2) + 6

= 2(4) + 6 + 6

= 8 + 12

= 20

Therefore, f(-2) = 20.

Step-by-step explanation:

Answer:

F(-2)=20

Step-by-step explanation:

You have to Substitute The X’s in your question with -2

This gives you: 2(-2)^2-3(-2)+6

2(-2)^2=2(4)=8 (Remember, A negative times a negative is a positive!)

-3(-2)=6

This leaves us with 8+6+6 which is equivalent to 20.

The value οf p that satisfies the prοpοrtiοn is 16.

Prοpοrtiοn is explained majοrly based οn ratiο and fractiοns. A fractiοn, represented in the fοrm οf a/b, while ratiο a:b, then a prοpοrtiοn states that twο ratiοs are equal. Here, a and b are any twο integers. The ratiο and prοpοrtiοn are key fοundatiοns tο understand the variοus cοncepts in mathematics as well as in science.

Prοpοrtiοn finds applicatiοn in sοlving many daily life prοblems such as in business while dealing with transactiοns οr while cοοking, etc. It establishes a relatiοn between twο οr mοre quantities and thus helps in their cοmparisοn.

Tο sοlve fοr p in the prοpοrtiοn:

10/p = 5/8

We can crοss-multiply tο οbtain:

5p = 80

Dividing bοth sides by 5, we get:

p = 16

Therefοre, the value οf p that satisfies the prοpοrtiοn is 16.

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Answer:

Step-by-step explanation:

To find the number of newborn babies weighing between 4 and 7 pounds, we need to calculate the z-scores for these two weights and use the normal distribution table.

For 4 pounds: z = (4 - 5.8) / 2.1 = -0.857

For 7 pounds: z = (7 - 5.8) / 2.1 = 0.571

Using the normal distribution table, the area between z = -0.857 and z = 0.571 is 0.5418.

So, the expected number of newborn babies weighing between 4 and 7 pounds is:

0.5418 x 1100 = 596 (rounded to the nearest whole number)

To find the number of newborn babies weighing less than 6 pounds, we need to calculate the z-score for 6 pounds and use the normal distribution table.

z = (6 - 5.8) / 2.1 = 0.095

Using the normal distribution table, the area to the left of z = 0.095 is 0.5375.

So, the expected number of newborn babies weighing less than 6 pounds is:

0.5375 x 1100 = 591 (rounded to the nearest whole number)

To find the number of newborn babies weighing more than 5 pounds, we need to calculate the z-score for 5 pounds and use the normal distribution table.

z = (5 - 5.8) / 2.1 = -0.381

Using the normal distribution table, the area to the right of z = -0.381 is 0.6499.

So, the expected number of newborn babies weighing more than 5 pounds is:

0.6499 x 1100 = 715 (rounded to the nearest whole number)

To find the number of newborn babies weighing between 5.8 and 10 pounds, we need to calculate the z-scores for these two weights and use the normal distribution table.

For 5.8 pounds: z = (5.8 - 5.8) / 2.1 = 0

For 10 pounds: z = (10 - 5.8) / 2.1 = 1.905

Using the normal distribution table, the area between z = 0 and z = 1.905 is 0.4713.

So, the expected number of newborn babies weighing between 5.8 and 10 pounds is:

0.4713 x 1100 = 518 (rounded to the nearest whole number)

If Colin's mom really drank 3 gallons of coffee, she would have filled her 12-fluid-ounce mug 32 times.

What is multiplication ?

Multiplication is a mathematical operation that involves combining groups of equal sizes to find a total quantity. It is represented by the symbol "×" or "⋅" and involves two or more factors. The result of a multiplication operation is called the product.

1 gallon is equal to 128 fluid ounces, so 3 gallons is equal to 3 x 128 = 384 fluid ounces.

To find out how many times Colin's mom would have filled her 12-fluid-ounce mug, we divide the total amount of coffee by the size of the mug:

384 fluid ounces ÷ 12 fluid ounces/mug = 32 mugs

Therefore, if Colin's mom really drank 3 gallons of coffee, she would have filled her 12-fluid-ounce mug 32 times.

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Answer:

The formula gives the area of a parallelogram:

Area = base x height

Substituting the given values, we get:

Area = 7.5 cm x 12.2 cm

Area = 91.5 cm^2

Therefore, the area of the parallelogram is 91.5 square centimeters.

Step-by-step explanation:

Answer:

91.5 sq cm

Step-by-step explanation:

A=bh=7.5·12.2=91.5

Answer:

The area of the rhombus CDEF is 32 units squared.

Step-by-step explanation:

The area of the rhombus CDEF is 24 square units. This can be determined by using the formula for finding the area of a rhombus which is A = 1/2 × d1 × d2 where d1 and d2 are the lengths of the diagonals of the rhombus. In this case, d1 is the distance from C to E and d2 is the distance from D to F. Calculating the lengths of the diagonals gives d1 = 8.9 and d2 = 8. Therefore, the area of the rhombus is 24 square units.

In the given problem, using central limit theorem,  the sample average score that corresponds to the 33rd percentile of all such sample averages is 100.66.

To solve this problem, we need to use the central limit theorem, which states that the distribution of sample averages is approximately normal, regardless of the distribution of the population, as long as the sample size is sufficiently large.

The mean of the sample means is equal to the population mean, which is 101. The standard deviation of the sample means, also known as the standard error of the mean, is equal to the population standard deviation divided by the square root of the sample size. In this case, the standard error of the mean is:

SE = 17 / sqrt(416) = 0.831

To find the sample average score that corresponds to the 33rd percentile of all such sample averages, we need to find the z-score that corresponds to the 33rd percentile and then use it to calculate the sample average score.

The z-score corresponding to the 33rd percentile is given by:

z = invNorm(0.33) = -0.439

where "invNorm" is the inverse normal distribution function.

We can use the formula for the z-score of a sample mean:

z = (x - μ) / SE

Rearranging this formula, we get:

x = z * SE + μ

Substituting the values we have, we get:

x = -0.439 * 0.831 + 101 = 100.66

Therefore, the sample average score that corresponds to the 33rd percentile of all such sample averages is 100.66.

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Answer:

Download 15 songs cost £22.50

Download 45 songs cost £67.50

Step-by-step explanation:

We know

Downloading 30 songs from a music website costs £45.

45 / 30 = £1.5 for 1 song

Download 1 song = £1.50

How much does it cost to download 15 songs?

1.5 x 15 = £22.50

How much does it cost to download 45 songs?

1.5 x 45 = £67.50

So,

Download 15 songs cost £22.50

Download 45 songs cost £67.50

Pythagorean theorem: This theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be expressed as a^2 + b^2 = c^2, where a and b are the lengths of the other two sides, and c is the length of the hypotenuse.

12² + 12² = x²

288= x²

x =16.98 = 17

Therefore , the solution of the given problem of equation comes out to be x = 100,003 is the value of x that fulfils the equation  log(x-3) = 5.

It is standard practise in mathematical formulas to employ a single variable term to ensure agreement between competing claims. Equations, which are mathematical statements, are used to demonstrate the equality of many academic expression numbers. Instead of dividing 12 into 2 parts in this instance, the normalise technique adds b + 6 to use the data from

y + 6.

Here,

By utilising the characteristics of logarithms, we can find x.

Beginning with the expression provided:

=>  log(x-3) = 5

It can be rewritten in exponential form as follows:

=> 10^5 = x - 3

When we simplify this solution, we obtain:

=> x = 10^5 + 3

We can calculate that 105 = 100,000 by using a calculator, so:

=> x = 100,000 + 3 = 100,003

As a result, x = 100,003 is the value of x that fulfils the equation

log(x-3) = 5.

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Answer:

Step-by-step explanation:

a. Journal Entries:

i. Budget Approval:

Debit: Appropriations - Salaries - Br. 800,000

Debit: Appropriations - Utilities - Br. 50,000

Debit: Appropriations - Equipment - Br. 85,000

Credit: Estimated Revenues - Property Taxes - Br. 850,000

Credit: Estimated Revenues - Licenses and Fees - Br. 70,000

Credit: Estimated Revenues - Investment Income - Br. 5,000

ii. Property Tax Levy:

Debit: Property Taxes Receivable - Br. 850,000

Credit: Property Taxes Revenue - Br. 850,000

iii. Collection of Revenues:

Debit: Cash - Br. 910,000

Credit: Property Taxes Receivable - Br. 840,000

Credit: Licenses and Fees Revenue - Br. 70,000

iv. Investment in CD:

Debit: Cash - Br. 50,000

Credit: Investments - Br. 50,000

v. Purchase of Police Sedans:

Debit: Equipment - Br. 80,000

Credit: Cash - Br. 80,000

vi. Payment for Police Sedans:

Debit: Equipment - Br. 2,000

Credit: Cash - Br. 82,000

vii. Payment of Salaries and Utility Bills:

Debit: Salaries Expense - Br. 770,000

Debit: Utilities Expense - Br. 45,000

Credit: Cash - Br. 815,000

b. Statement of Revenues, Expenditures, and Changes in Fund Balance:

Revenues:

Property Taxes Revenue - Br. 850,000

Licenses and Fees Revenue - Br. 70,000

Investment Income - Br. 2,000

Total Revenues - Br. 922,000

Expenditures:

Salaries Expense - Br. 800,000

Utilities Expense - Br. 50,000

Capital Outlay - Br. 82,000

Total Expenditures - Br. 932,000

Net Change in Fund Balance:

Revenues - Expenditures - Br. 10,000

Fund Balance:

Beginning Fund Balance - Br. 50,000

Net Change in Fund Balance - Br. 10,000

Ending Fund Balance - Br. 40,000

c. Balance Sheet:

Assets:

Cash - Br. 58,000

Investments - Br. 50,000

Property Taxes Receivable - Br. 10,000

Equipment - Br. 85,000

Total Assets - Br. 203,000

Liabilities:

None

Fund Balance:

Unassigned - Br. 40,000

Total Liabilities and Fund Balance - Br. 203,000

No, Kiran is incorrect. -2 is actually greater than -5, so the correct statement is -5 < -2.

A number line is an image of numbers plotted on a straight line, either horizontally or vertically. We can compare numbers and execute simple arithmetic operations on them easily by writing the numbers down on a number line. The starting point of a number line is often regarded as zero (0). The numbers to the left of zero are all negative whereas the numbers on the right of zero are all positive. As a result, we may argue that on a number line, the value of numbers grows as we approach towards the right. The numbers on the right are thus greater than the ones on the left, according to this statement.

No, Kiran is incorrect. -2 is actually greater than -5, so the correct statement is -5 < -2.

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Step 1: Convert zeros to factors of the polynomial.

If "k" is a zero, then (x-k) is a factor of the polynomial.

The allows us to convert zeros of -3, -1, and 2 into the factors (x+3)(x+1)(x-2).

So right now we have y= (x+3)(x+1)(x-2), but there's a missing piece, the leading coefficient of "a", so really we currently have

   y = a · (x+3)(x+1)(x-2)

This is where the point (3,5) comes in.  We need to substitute x=3 and y=5 into the equation above to find a.

        5 = a · (3+3)(3+1)(3-2)

        5 = a · (6)(4)(1)

        5 = a · 24

   5/24 = a

That's the full function:

    f(x) = 5/24 (x+3)(x+1)(x-2)

Answer:

[tex]f(x)=\dfrac{5}{24}x^3+\dfrac{5}{12}x^2-\dfrac{25}{24}x-\dfrac{5}{4}[/tex]

Step-by-step explanation:

The zeros of a polynomial f(x) are the values of x which satisfy f(x) = 0.

If the polynomial function is degree 3 with 3 zeros, in factored form it can be written as f(x) = k(x - r₁)(x - r₂)(x - r₃) where r are the zeros and k ≠ 0.

Substitute the given zeros -3, -1 and 2 into the formula:

[tex]\implies f(x)=k(x-(-3))(x-(-1))(x-2)[/tex]

[tex]\implies f(x)=k(x+3)(x+1)(x-2)[/tex]

To find the value of k, substitute the given point (3, 5) into the function:

[tex]\begin{aligned}\implies f(3)&=5\\k(3+3)(3+1)(3-2)&=5\\k(6)(4)(1)&=5\\24k&=5\\k&=\dfrac{5}{24}\end{aligned}[/tex]

Therefore, the factored form of the polynomial function is:

[tex]\implies f(x)=\dfrac{5}{24}(x+3)(x+1)(x-2)[/tex]

To write in standard form, expand and distribute:

[tex]\implies f(x)=\dfrac{5}{24}(x^2+4x+3)(x-2)[/tex]

[tex]\implies f(x)=\dfrac{5}{24}(x^3+2x^2-5x-6)[/tex]

[tex]\implies f(x)=\dfrac{5}{24}x^3+\dfrac{10}{24}x^2-\dfrac{25}{24}x-\dfrac{30}{24}[/tex]

[tex]\implies f(x)=\dfrac{5}{24}x^3+\dfrac{5}{12}x^2-\dfrac{25}{24}x-\dfrac{5}{4}[/tex]

The parametric equations for the curve are X = 4 sin(5t), Y = -48 sin²(5t) + 8 sin(5t) + 45

A parametric equation is a set of equations that express the coordinates of a point as functions of one or more parameters. These equations define a curve or a surface in a space and are commonly used in mathematics, physics, engineering, and computer graphics to model and analyze various phenomena.

Parametric equations can be useful in many applications, such as modeling the motion of particles, designing curves and surfaces, and creating computer animations. They also allow for the visualization of complex shapes and curves that may be difficult to describe using traditional equations.

To find the parametric equation of the given curve, we need to express the coordinates x and y in terms of a parameter t. We can use the trigonometric identities to simplify the given equations and get:

X = 4 sin(5t)

Y = -48 sin²(5t) + 8 sin(5t) - 3

Let's start with Y. We can use the identity sin²(θ) + cos²(θ) = 1 to rewrite -48 sin²(5t) as -48 (1 - cos²(5t)) = -48 cos²(5t) + 48. Substituting this into the equation for Y, we get:

Y = -48 cos²(5t) + 8 sin(5t) - 3

Now, we can use another trigonometric identity, sin²(θ) + cos²(θ) = 1, to rewrite cos²(5t) as 1 - sin²(5t). Substituting this into the equation for Y, we get:

Y = -48 (1 - sin²(5t)) + 8 sin(5t) - 3

Y = -48 sin²(5t) + 8 sin(5t) + 45

So, the parametric equations for the curve are:

X = 4 sin(5t)

Y = -48 sin²(5t) + 8 sin(5t) + 45

This is the final answer.

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