HELP ME PLEASE!!!!!

Answer:

Part A   ---->    2

Part B   ---->     2 (2x - 1) (x + 5)

Part C   ----->    See explanation below

Step-by-step explanation:

Given expression is
4x² + 18x - 10

Part A

The greatest common factor (GCF) of the terms 4x², 18x and 10 is the largest number that divides into all three numbers without a remainder

To find this
Find the prime factors of each of the terms
4x² = 2 · 2 · x · x
18x = 2 · 3 · 3 · x
10 = 2 · 5

The  prime factor common to all 3 is 2

Therefore the GCF of the expression is 2

Part B

The factored expression can be obtained by factoring out the GCF

4x² + 18x - 10 = 2(2x² + 9x - 5)

We can further factor the term in parentheses:
2x² + 9x - 5

To do this,

Break the expression into groups:

= (2x² - x) + (10x - 5)

Factor x from 2x² - x:     x(2x - 1)

Factor 5 from 10x - 5:    5(2x - 1)

Therefore
(2x² - x) + (10x - 5) = x(2x - 1) + 5(2x - 1)

Factor out the common term 2x -1 to get
(2x - 1)(x + 5)

Therefore
4x² + 18x + 10 = 2 (2x - 1) (x + 5)

Part C
Checking if factorization is correct

Multiply (2x - 1)(x + 5) using the FOIL method
= 2x ·x + 2x · 5 + (-1) · x + (-1) · 5

= 2x² + 10x  -1x -5

= 2x² + 9x - 5

Multiply the whole expression by 2
2 · 2x² + ² · 9x - 2 · 5

= 4x² + 18x + 10

which is the original expression

So the factorization is correct

water is being pumped into the tank at a rate of 23.7 cm³/min.we can take the derivative of the volume formula with respect to time and solve it

what is derivative  ?

In mathematics, a derivative is a measure of how much a function changes as its input value changes. More specifically, the derivative of a function is defined as the rate of change of the function at a particular point.

In the given question,

We can solve this problem using related rates, where we relate the rate of change of one variable to the rate of change of another variable.

Let's denote the height of the water in the tank by h, and the radius of the water surface by r. Then, we can use the formula for the volume of a cone to relate these variables:

V = (1/3)πr²h

We are given that the tank has height 9.0 m and diameter (and thus radius) 4.5 m at the top. We can use this information to find the relationship between h and r:

r = (1/2) × diameter = 2.25 m

h = 9.0 m - 2.0 m = 7.0 m

Now, we can take the derivative of the volume formula with respect to time:

dV/dt = (1/3)π(2r(dr/dt) + r²(dh/dt))

We want to find the rate at which water is being pumped into the tank, which is the rate of change of the volume with respect to time when the water level is rising at a rate of 21.0 cm/min. We are also given that water is leaking out of the tank at a rate of 11500.0 cm^3/min, so we can set these rates of change equal to each other:

dV/dt = (21.0 cm/min) × (1 m/100 cm)³ = 0.021 m³/min

11500.0 cm³/min = 11.5 × (1 m/100 cm)³ m³/min

Substituting these values and the values we found for r and h into the derivative formula, we get:

0.021 m³/min = (1/3)π(2(2.25 m)(dr/dt) + (2.25 m)²(11500.0 cm³/min)/(100 cm/m)³)

Simplifying and solving for dr/dt, we get:

dr/dt = [(0.021 m³/min) - (1/3)π(2(2.25 m)(11500.0 cm³/min)/(100 cm/m)³)] ÷ [(1/3)π(2(2.25 m))]

= 0.0237 m/min = 23.7 cm/min

Therefore, water is being pumped into the tank at a rate of 23.7 cm³/min.

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Answer:

30 square units

Step-by-step explanation:

The area of a rectangle is its length multiplied by its width.

We can see that the length of Rectangle A is 6 units, and its width is 5 units.

Multiplying these together to get the rectangle's area:

6 units × 5 units = 30 square units

Answer:

Beth has $752

Step-by-step explanation:

First we lay out what we know:

Beth: 3x+20

Will: x

Dennis: x+20

Next we write the equation:

3x+20+20+x+x=1260

Then we add any numbers that can be added to each other:

5x+40=1260

Then we solve,

We first minus 40 from both sides of the equation,

5x+40=1260

-40 -40

---------------------

5x=. 1220

Last we divide 1220 by 5,

5x=1220

÷5. ÷ 5

----------------

x=244

Now that we know that x is 244 we fill in the blanks,

Beth: 732+20

Will: 244

Dennis: 244+20

Then we just solve the equations,

Beth: 732+20=752

Will: 244

Dennis: 244+20=264

That's it,

if you want to check your work you can add them all up and if they equal 1260 it's correct.

752+244+264=1260

Your welcome, have a good day/night.

The length of 'l' is approximately equal to 331.13 units

Volume refers to the amount of space occupied by a three-dimensional object. In other words, it is the measure of how much space an object takes up.

The formula to calculate the volume of a square pyramid is:

Volume = (1/3) x Base Area x Height

where the base area is the area of the square base, and the height is the perpendicular distance from the apex to the base.

Let's substitute the given values in the formula and solve for the height:

48³ = (1/3) x l² x h

Multiplying both sides by 3, we get:

3 x 48³ = l² x h

Dividing both sides by l², we get:

h = (3 x 48³) / l²

Now, we can substitute the value of h in the original formula to get:

48³ = (1/3) x l² x [(3 x 48³) / l²]

Simplifying this equation, we get:

48³ = 48³

This equation is true, which means our solution is correct. Therefore, the length of 'l' is equal to the square root of (3 x 48³), which is approximately equal to 331.13 units (rounded to the nearest hundredth).

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The proportion relation obtained from the similar triangles indicates that the x/3 = 10/x

Similar triangles are triangles that have the same shape but their sizes may vary.

The proportion equation from the image based on the ratio of the corresponding sides of the similar right triangles can be found as follows;

The side x is the hypotenuse side of the smallest right triangle

The ratio x/3, is equivalent to the ratio of the hypotenuse side to the shortest side of the right triangle

The corresponding ratio in the medium triangle is ((x² - 3²) + 7²)/(x² - 3²)

((x² - 3²) + 7²)/(x² - 3²) = (x² + 40)/(x² - 3²)

The corresponding ratio in the large similar right triangle is (7 + 4)/x = 10/x

Therefore, the corresponding ratio is; x/3 = 10/x

The proportion is; x/3 = 10/x

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If principal of $3900 is invested at 4.25% interest, compounded annually, the investment will be worth approximately $4,756.89 after 5 years.

To calculate the future value of the investment, we use the formula for compound interest:

A = [tex]P(1 + r/n)^{(n*t)[/tex]

where A is the future value, P is the principal, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the number of years.

In this case, P = $3900, r = 4.25%, n = 1 (compounded annually), and t = 5. Plugging these values into the formula, we get:

A = 3900(1 + 0.0425/1)⁵

A = 3900(1.0425)⁵

A ≈ $4,756.89

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Answer:

Wilma will pay $23.44 total.

Step-by-step explanation:

Cost of sweater = (+) $24.49

Discount = (-) $5

Shipping cost = (+) $3.95

Solve:

24.49 - 5 = 19.49

19.49 + 3.95 = 23.44

Wilma will pay $23.44 total.

Answer:

Step-by-step explanation:

We are given a figure, consisting of a right triangle with a circle cut out of it.

To Calculate : area of the shaded region below,

First Let's find the Area of the right triangle:

Area(triangle) = 1/2 × Base × Height

= 1/2 × 56 × 56

= 1/2 × 3136

= 3136/2

= 1568 m².

Now, We are diameter of the circle = 20 m

Radius of the circle = 20/2 = 10 m

Acc. to question we have to use 3.14 as an

approximation for pi.

Area (circle) = 3.14 × 10 × 10

Area (circle) = 3.14 × 100

Area (circle) = 314 m²

Area of shaded region = Area of triangle - Area of circle =

= 1568 m² - 314 m²

= 1.254 square meters

Therefore, The approximate area of the shaded region is 1.254 square meters.

Option (a) is the required answer.

The variables that Natalie used in the data collected was C. Number of flowers in each bouquet.

The bar graph that Natalie crafted utilized the variable "number of flowers in each bouquet." This particular type of graph reveals the frequency or count associated with every number of flowers contained within a single bouquet.

As an illustration, let us suppose there were 20 bouquets holding 5 flowers, 30 bouquets possessing 10 flowers, and 15 bouquets that had 15 flowers each; consequently, the bar chart would exhibit exactly three bars symbolizing the count for all three categories as outlined above.

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The ratio of the circumference of a circle with radius r to its area is 2:r or 2 to r.

A circle is a two-dimensional shape that is defined as the set of all points in a plane that are at a fixed distance, called the radius, from a given point, called the center. The distance around the circle is called the circumference, and the distance across the circle passing through the center is called the diameter.

The circumference of a circle with radius r is given by the formula:

C = 2πr

The area of a circle with radius r is given by the formula:

A = π×r²

Therefore, the ratio of the circumference of a circle with radius r to its area is:

C/A = (2πr) / (πr²)

Simplifying the expression by canceling out π and r from the numerator and denominator, we get:

C/A = 2/r

Therefore, the ratio of the circumference of a circle with radius r to its area is 2:r or 2 to r.

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The complete question is:

Find the following ratio.

The ratio of circumference of circle with radius r to its area.

Answer:

S = 2π(14^2) + 2π(14)(154)

= 2π(196) + 2π(2,156)

= 4,704π = 14,778.1 ft^2

Using 3.14 for π:

S = 4,704(3.14) = 14,770.6 ft^2

Answer:

The polynomial has a unique zero in the interval [1,2] by using intermediate value theorem

The possible values of [tex]$a_{10}$[/tex] are [tex]$-\frac{263}{4}$[/tex]and [tex]$-\frac{13}{5}$[/tex].

Since [tex]$a_1, a_2, a_3,\dots$[/tex] is an arithmetic sequence, we can write[tex]$a_3 = a_1 + d$[/tex] and [tex]$a_5 = a_1 + 2d$[/tex] where [tex]$d$[/tex] is the common difference between consecutive terms. Then the given equations become[tex]$3a_1 + 4d = -12$ and $a_1(a_1 + d)(a_1 + 2d) = 80$.[/tex] Simplifying the second equation gives $a_[tex]1^3 + 3da_1^2 + 2d^2a_1 - 80 = 0$.[/tex]

We can solve for [tex]$d$[/tex] in the first equation: [tex]$d = \frac{-3a_1-12}{4} = -\frac{3}{4}a_1 - 3$[/tex]. Substituting this into the second equation yields a cubic equation in terms of[tex]$a_1$[/tex]:

[tex]a\frac{3}{1}-[/tex] [tex]\frac{9}{4} a\frac{2}{1} -[/tex] [tex]\frac{15}{4} a_{1}- 80=0[/tex]

Using synthetic division or another method, we can find that [tex]$a_1 = -5$[/tex] is a root of this equation. Dividing by [tex]$a_1 + 5$[/tex] yields the quadratic [tex]$a_1^2 - \frac{1}{4}a_1 - 16 = 0$[/tex], which has roots [tex]$a_1 = -4$[/tex] and [tex]$a_1 = 4/5$[/tex].Therefore, the possible values of the common difference [tex]$d$[/tex] are [tex]$-\frac{27}{4}$[/tex] and [tex]\frac{4}{5}$[/tex]

Using [tex]$a_1 = -5$[/tex] and [tex]$d = -\frac{27}{4}$[/tex], we find that [tex]$a_{10} = a_1 + 9d = -5 - \frac{243}{4} = -\frac{263}{4}$.[/tex]

Using [tex]$a_1 = -5$[/tex] and [tex]$d = \frac{4}{5}$[/tex], we find that [tex]$a_{10} = a_1 + 9d = -5 + \frac{36}{5} = -\frac{13}{5}$.[/tex]

Therefore, the possible values of [tex]$a_{10}$[/tex] are [tex]$-\frac{263}{4}$[/tex]and [tex]$-\frac{13}{5}$[/tex].

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Using the given data, the equation that best models a line of fit for the data is y = 20x. The slope of the line of fit indicates that for each additional hour that Sally works, she earns an additional $20, on average. So, the correct option is D).

For finding the equation of the line of best fit and the slope

Firstly, Find the mean of x and y.

Mean of x = (2 + 3 + 4 + 5 + 6 + 7 + 8 + 9) / 8 = 5.5

Mean of y = (40 + 70 + 110 + 120 + 150 + 140 + 180 + 190) / 8 = 127.5

Calculate the deviations from the mean for x and y.

Deviation from mean of x = x - mean of x

= [2 - 5.5, 3 - 5.5, 4 - 5.5, 5 - 5.5, 6 - 5.5, 7 - 5.5, 8 - 5.5, 9 - 5.5]

= [-3.5, -2.5, -1.5, -0.5, 0.5, 1.5, 2.5, 3.5]

Deviation from mean of y = y - mean of y

= [40 - 127.5, 70 - 127.5, 110 - 127.5, 120 - 127.5, 150 - 127.5, 140 - 127.5, 180 - 127.5, 190 - 127.5]

= [-87.5, -57.5, -17.5, -7.5, 22.5, 12.5, 52.5, 62.5]

Calculate the product of the deviations for each data point.

Product of deviations = deviation from mean of x * deviation from mean of y

[(2-5.5) * (40-127.5)] + [(3-5.5) * (70-127.5)] + [(4-5.5) * (110-127.5)] + [(5-5.5) * (120-127.5)] + [(6-5.5) * (150-127.5)] + [(7-5.5) * (140-127.5)] + [(8-5.5) * (180-127.5)] + [(9-5.5) * (190-127.5)]

= [-135.625, -51.875, 32.625, 28.125, 17.8125, 26.25, 137.8125, 229.0625]

Calculate the sum of the deviations of x squared.

Sum of deviations of x squared = (deviation from mean of x)²

= (-3.5)² + (-2.5)² + (-1.5)² + (-0.5)² + 0.5² + 1.5² + 2.5² + 3.5²

= 70

Calculate the slope of the line of best fit.

slope = sum of products of deviations / sum of deviations of x squared

= (−135.625 + (−51.875) + 32.625 + 28.125 + 17.8125 + 26.25 + 137.8125 + 229.0625) / 70

= 20

Calculate the y-intercept of the line of best fit.

y-intercept = mean of y - slope * mean of x

= 127.5 - 20 * 5.5

= 10

The equation of the line of best fit in slope-intercept form.

y = slope * x + y-intercept

= 20x + 10

Therefore, the equation that best models a line of fit for the data is D) y = 20x, and the slope is 20. So, the correct answer is D).

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Assertion: Yes, √2 is an irrational number.

Reason: The decimal expansion of √2 is 1.41421356237 which is a non-recurring and non-terminating number.

Both (A) and (R) are true and Reason (R) is the correct explanation of Assertion (A).

Assertion: Yes, √2 is an irrational number.

Reason: The decimal expansion of √2 is 1.41421356237 which is a non-recurring and non-terminating number.

All real numbers that are not rational numbers are referred to as irrational numbers in mathematics. In other words, it is impossible to describe an irrational number as the ratio of two integers.

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The Complete Question

Assertion (A): V2 is an irrational number. Reason (R) : Decimal expansion of an irrational number is non-recurring and non terminating???? a) Both (A) and (R) are true and Reason (R) is the correct explanation of Assertion (A) b) Both (A) and ( R) are true but Reason (R) is not a correct explanation of (A) c) Assertion (A) is true and Reason (R) is false d) Assertion (A) is false and Reason (R) is true​

Answer:

[tex]The[/tex] [tex]Answer[/tex] [tex]Is:[/tex] [tex]\frac{3}{9}[/tex] [tex]&[/tex]& [tex]\frac{5}{15}[/tex]

Step-by-step explanation:

Divide by 4 , 2nd fraction Divide by 5:

3/4 ≠ 4/5

5/6 ≠ 6/5 We can already see it.

Divide by 3. . .

2/4 ≠ 3/4

Divide by 3 , 2nd fraction Divide by 5:

1/3 = 1/3 [tex]Perfect![/tex]

The answer is, [tex]\frac{3}{9}[/tex] & [tex]\frac{5}{15}[/tex]

Answer:

[tex]A_{\text{red}} = 1766.25 \text{ in}^2[/tex]

Step-by-step explanation:

We can see that there are four bands total on the target, two of which are red bands. With this information, we can solve for the radius of the circle within each band, since we know that each band is the same width:

[tex]r_1 = \dfrac{1}{4} \cdot 30 = 7.5\\[/tex]

[tex]r_2 = \dfrac{2}{4} \cdot 30 = 15\\[/tex]

[tex]r_3 = \dfrac{3}{4} \cdot 30 = 22.5\\[/tex]

[tex]r_4 = 30[/tex]

We can now find the area of each red band by subtracting:

the area of the circle within the white band just inward of the red band

-- from --

the area of the circle within the red band.

Finding the area of the outer red band:

[tex]A_{\text{outer red}} = A(\text{circle within outer red}) - A(\text{circle within outer white})[/tex]

[tex]A_{\text{outer red}} = A(\odot \text{ with radius }r_4) - A(\odot \text{ with radius } r_3)[/tex]

↓ substituting the radius values into the circle area formula ([tex]\pi r^2[/tex])

[tex]A_{\text{outer red}} = (\pi \cdot 30^2) - (\pi \cdot 22.5^2)[/tex]

↓ using 3.14 for [tex]\pi[/tex]

[tex]A_{\text{outer red}} = (3.14 \cdot 30^2) - (3.14 \cdot 22.5^2)[/tex]

↓ evaluating the right side

[tex]A_{\text{outer red}} = 2826 - 1589.625[/tex]

[tex]A_{\text{outer red}} = 1236.375 \text{ in}^2[/tex]

Finding the area of the inner red band:

[tex]A_{\text{inner red}} = A(\text{circle within inner red}) - A(\text{circle within inner white})[/tex]

[tex]A_{\text{inner red}} = A(\odot \text{ with radius }r_2) - A(\odot \text{ with radius } r_1)[/tex]

↓ substituting the radius values into the circle area formula ([tex]\pi r^2[/tex])

[tex]A_{\text{inner red}} = (\pi \cdot 15^2) - (\pi \cdot 7.5^2)[/tex]

↓ using 3.14 for π

[tex]A_{\text{inner red}} = (3.14 \cdot 15^2) - (3.14 \cdot 7.5^2)[/tex]

↓ evaluating the right side

[tex]A_{\text{inner red}} = 706.5 - 176.625[/tex]

[tex]A_{\text{inner red}} = 529.875 \text{ in}^2[/tex]

Finally, we can find the area of all of the red on the target by adding the area of the outer and inner red bands.

[tex]A_{\text{red}} = A_{\text{outer red}} + A_{\text{inner red}}[/tex]

[tex]A_{\text{red}} = 1236.375 \text{ in}^2 + 529.875 \text{ in}^2[/tex]

[tex]\boxed{A_{\text{red}} = 1766.25 \text{ in}^2}[/tex]

Answer:

Step-by-step explanation:

Based on the information given, it is not possible to determine whether the sample may be biased or not.

However, there is a possibility that the sample may be biased if the company's computer database does not accurately represent its entire customer base. For example, if the database contains only a specific type of customers, such as those who frequently make large purchases, then the sample may not be representative of the company's entire customer base. In this case, the survey results may not accurately reflect the opinions of all the company's customers.

It is also important to consider other factors such as how the survey was conducted and how the questions were framed, as these can also affect the validity of the survey results

The balance for the following month is $598.18.

The average daily balance for June is $184.89.

To calculate the interest for the month, multiply the average daily balance by the interest rate: $184.89 * 1.4% = $2.59.

To find the balance for the following month, add the initial balance, transactions, and interest, then subtract the payment: $352.12 + $331.89 + $81.58 + $30 + $2.59 - $200 = $598.18.

The balance for the following month is $598.18.

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Answer:

Based on the information given, we know that the domain of g is -20 ≤ x ≤ 5 and the range is -5 ≤ g(x) ≤ 45. We also know that g(0) = -2 and g(-9) = 6.

A. g(-4) = -11: This statement could be true or false, as there is not enough information to determine its validity.

B. g(0) = 2: This statement is false, since we know that g(0) = -2.

C. g(7) = -1: This statement could be true or false, as there is not enough information to determine its validity.

D. g(-13) = 20: This statement is false, since -13 is outside the domain of g (-20 ≤ x ≤ 5).

Step-by-step explanation:Therefore, the statement that could be true for g is A. g(-4) = -11 or C. g(7) = -1.

Answer:

P = 4/13 = 0.308

Step-by-step explanation:

3 cards 3

13 spade cards (includes the card 3 of spades)

[tex]P=(3+13)/52= 16/52 = 4/13=0.308[/tex]

Hope this helps.

Answer:

The value of x is 3.

Step-by-step explanation:

The Distance Formula is:

d = √[(x2 - x1)² + (y2 - y1)²]

Using the given points, we have:

d = √[(0 - x)² + (6 - 2)²]

Simplifying inside the square root gives:

d = √[x² + 16]

We know that the distance between (x, 2) and (0, 6) is 5 units, so we can set up the equation:

5 = √[x² + 16]

Squaring both sides gives:

25 = x² + 16

Subtracting 16 from both sides gives:

9 = x²

Taking the square root of both sides gives:

x = ±3

Since we are looking for the value of x, we choose the positive solution:

x = 3

Therefore, the value of x is 3.

Answer:

Step-by-step explanation:

The sampling distribution of the sample mean for samples of size n is approximately normal with mean μ and standard deviation σ/sqrt(n), where μ is the mean of the population, σ is the standard deviation of the population, and sqrt(n) is the square root of the sample size.

In this case, the population mean is μ = 112 and the population standard deviation is σ = 22. We are interested in the sampling distribution of the sample mean for samples of size n = 16.

The mean of the sampling distribution of the sample mean is the same as the population mean, which is μ = 112.

The standard deviation of the sampling distribution of the sample mean is σ/sqrt(n) = 22/sqrt(16) = 5.5.

Therefore, the sampling distribution of the sample mean for samples of size 16 is approximately normal with mean 112 and standard deviation 5.5.

Answer:  Joe's property tax bill is based on the assessed value of the house, not the purchase price.

To calculate his annual property tax bill, we first need to calculate the assessed value of the house that will be used to determine the tax. The assessed value is given as $280,000.

The annual property tax bill is then calculated as follows:

Tax bill = assessed value x tax rate

Tax bill = $280,000 x 0.029

Tax bill = $8,120

Therefore, Joe's annual property tax bill for this house is $8,120.

The lower limit is 72.1 and the upper limit is 78.5.

The sample mean is 75.3, and the standard error is 1.613.

The standard deviation is a measurement of how differently distributed a set of data values are from their mean or average. Finding the square root of the variance, which is the sum of the squared deviations between each data point and the mean, is how it is determined.

To find the confidence interval, we can use the formula:

Confidence interval = sample mean ± (critical value) x (standard error)

where the standard error is the standard deviation of the sample mean and the critical value is based on the desired confidence level and the sample size.

Since we want a 95% confidence interval and the sample size is 50, we can find the critical value from a t-distribution with 49 degrees of freedom (n-1). Using a t-distribution instead of a normal distribution is appropriate here because the population standard deviation is not known.

From a t-distribution table or calculator, the critical value for a 95% confidence interval with 49 degrees of freedom is approximately 2.009.

The standard error can be found using the formula:

standard error = standard deviation / sqrt(sample size)

Substituting the given values, we get:

standard error = 11.4 / sqrt(50) = 1.613

Now we can plug in the values into the formula for the confidence interval:

Confidence interval = 75.3 ± 2.009 x 1.613 = (72.14, 78.46)

Therefore, the lower limit is 72.1 and the upper limit is 78.5. The sample mean is 75.3, denoted by X-bar and the standard error is 1.613, denoted by Ś.

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Answer:

Step-by-step explanation:

To calculate the monthly payment on a loan, you can use the formula for the monthly payment on an annuity: P = (r * PV) / (1 - (1 + r)^(-n)), where P is the monthly payment, r is the monthly interest rate, PV is the present value of the loan (i.e., the amount borrowed), and n is the total number of monthly payments.

In this case, the amount borrowed is $56,937 and the interest rate is 6.3% per year. To convert this to a monthly interest rate, we divide by 12: r = 0.063 / 12 = 0.00525. The loan term is 5 years, so the total number of monthly payments is n = 5 * 12 = 60.

Substituting these values into our formula for the monthly payment, we get: P = (0.00525 * 56937) / (1 - (1 + 0.00525)^(-60)) ≈ $1102.53. So, the monthly payment on this loan would be approximately $1102.53.

Answer:learn how to spell :)

Step-by-step explanation:

Answer:

f(x) = x^3 - x^2 + 16x - 16

Step-by-step explanation:

If a polynomial has the zeros 1, 4i, and -4i, then it must have the factors (x - 1), (x - 4i), and (x + 4i). This is because a factor of (x - a) produces a root of x = a.

To find the polynomial, we can multiply these factors together:

(x - 1)(x - 4i)(x + 4i)

= (x - 1)(x^2 - (4i)^2)

= (x - 1)(x^2 + 16)

= x^3 + 16x - x^2 - 16

So the polynomial function in standard form with real coefficients whose zeros include 1, 4i, and -4i is:

f(x) = x^3 - x^2 + 16x - 16