The expression for the total number of cups Alison buys, in terms of p and b, is 12p + 18b.
The total number of cups Alison buys can be expressed in terms of the number of packs (p) and boxes (b) purchased.
In each pack, there are 12 cups,
so the total number of cups from packs would be 12p.
Similarly, in each box, there are 18 cups,
so the total number of cups from boxes would be \: 18b.
To find the total number of cups, we sum the cups from packs and cups from boxes:
Total cups = 12p + 18b
Therefore, the expression for the total number of cups Alison buys, in terms of p and b, is 12p + 18b.
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Captain’s Autos sells 22 used cars on Monday, and 18 cars on Tuesday. This was 25% of the number of sales for the week. How many cars did they sell altogether of the week?
The number of cars Captain's Autos sold in total that week is 160.
What is the unitary method?The unitary method is a method for solving a problem by the first value of a single unit and then finding the value by multiplying the single value. Unitary method is a technique by which we find the value of a single unit from the value of multiple devices and the value of more than one unit from the value of a single unit. It is a method that we use for most of the calculations in math.
We are given that:
Number of used cars on Monday = 22Number of cars on Tuesday = 18Now,
Let's call the total number of cars sold during the week "x".
We know that the number of cars sold on Monday and Tuesday is 25% of the total number of cars sold during the week. So we can write:
[tex]\sf 22 + 18 = 0.25x[/tex]
Simplifying, we get:
[tex]\sf 40 = 0.25x[/tex]
Dividing both sides by 0.25, we get:
[tex]\bold{x = 160}[/tex]
Hence, by the unitary method the answer will be 160.
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which image is the translation of triangle ABC given by the translation rule (x,y) (x-2,y+3)
The coordinates of the image of triangle ABC would be A (-1, -1), B (3, 2), and C (4, -6) after translation, and if the original coordinates are A(-3,2) B(1,5) C(2,-3).
We know that,
A point is transformed when it is moved from where it was originally to a new location. Translation, rotation, reflection, and dilation are examples of different transformations.
Let assume A(-3,2) B(1,5) and C (2,-3) are original coordinates
The following translation is being used: (x, y) ⇒ (x + 2, y - 3)
First, take your x-coordinates and add them by two because it is the translation utilized to solve for your x-coordinate.
You would receive the following for yours x's: A (-1, y ) B (3, y) C (4, y)
Next, remove three from each of your y-coordinates because it is the translation being utilized to solve for your y-coordinate.
You would receive the following for your y's: A (x, -1) B (x, 2) C (x, -6)
Finally, you would take each one and put it together, or piece it together.
Hence, the coordinates of the image of triangle ABC after translation would be A (-1, -1), B (3, 2), and C (4, -6).
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A data set is normally distributed with a mean of 27 and a standard deviation of 3.5. About what percent of the data is greater than 34?
Answer:
Approximately 2.5% of the data is greater than 34.
Step-by-step explanation:
To solve this problem, we need to use the empirical rule (also known as the 68-95-99.7 rule) for a normal distribution. This rule states that about 68% of values lie within 1 standard deviation of the mean, about 95% of the values lie within 2 standard deviations of the mean, and about 99.7% of the values lie within 3 standard deviations of the mean.
The mean of the dataset is 27, and the standard deviation is 3.5.
34 is exactly 2 standard deviations away from the mean (since 27 + 2*3.5 = 34). According to the empirical rule, about 95% of the data falls within this range. This means that about 5% of the data is outside of this range.
Since the normal distribution is symmetrical, the data outside of 2 standard deviations is equally split between values that are too large and too small. Hence, about half of this 5%, or 2.5%, is greater than 34.
Therefore, approximately 2.5% of the data is greater than 34.
Housing prices in a small town are normally distributed with a mean of
131,000 and a standard deviation of 8,000
. Use the empirical rule to complete the following statement.
About 95% of the housing Prices are between (µ - 2σ) and (µ + 2σ).About 99.7% of the housing prices are between (µ - 3σ) and (µ + 3σ)
Given that housing prices in a small town are normally distributed with a mean of µ. We are to use the empirical rule to complete the following statement.
The empirical rule states that in a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, approximately 95% of the data falls within two standard deviations of the mean,
and approximately 99.7% of the data falls within three standard deviations of the mean.Since we do not have information about the standard deviation of the housing prices,
we cannot provide exact values for the empirical rule. However, we can make some general statements:
About 68% of the housing prices are between (µ - σ) and (µ + σ).
About 95% of the housing prices are between (µ - 2σ) and (µ + 2σ).About 99.7% of the housing prices are between (µ - 3σ) and (µ + 3σ)
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Arrange the following temperatures in ascending order and descending order.
a) 37°C, -15°C, 16°C, -12°C, 0°C, 96°C, -73°C
b) 20°C, -1°C -15°C 0°C, -7°C, 23°C, -36°C.
Answer:
a) -73°C, -15°C, -12°C, 0°C, 16°C, 37°C, 96°C
a) 96°C, 37°C, 16°C, 0°C, -12°C, -15°C, -73°C
b) -36°C, -15°C, -7°C, -1°C, 0°C, 20°C, 23°C
b) 23°C, 20°C, 0°C, -1°C, -7°C, -15°C, -36°C
A bakery offers a sale price of $3.00 for 6 muffins. What is the price per dozen?
Answer:
$6.00
Step-by-step explanation:
In order y find the cost of one muffin, we divide the price of 6 muffins by 6.
$3.00 / 6 = $0.50
Therefore, the cost of one muffin is $0.50.
To find the price per dozen, we multiply the cost of one muffin by 12.
$0.50 * 12 = $6.00
Therefore, the price per dozen muffins is $6.00.
Which is the graph of f(x) = (x-1)(x + 4)?
O
2
AV
2
--2-
O
6
O
Answer:
The bottom one
Step-by-step explanation:
Notice that the graph is an upward opening parabola and passes through the x-axis at the x-intercepts x=1 and x=-4. These are our zeroes that allow f(x)=0 by the Zero Product Property.
Numbers and operations
show work on paper if possible :)
Answer:
B. $5.99
Step-by-step explanation:
Let's start by finding the total cost of the sodas. We know that there were 5 sodas and each cost $0.75, so the total cost of the sodas is:
5 * $0.75 = $3.75
Next, we need to subtract the cost of the sodas from the total cost before tax to find the cost of the pizzas:
$33.70 - $3.75 = $29.95
Finally, we can divide the cost of the pizzas by the number of pizzas to find the cost of each pizza:
$29.95 ÷ 5 = $5.99
Therefore, the cost of each pizza pie was $5.99. Answer choice B is correct.
A gazebo in the shape of a regular hexagon (six sides) has side length s and apothem a. If the cost per square unit of flooring is c, express the total cost T of the floor algebraically.
The total cost T of the floor algebraically is T = c x (3√3/2) x s²
We have,
To express the total cost T of the floor of the hexagonal gazebo algebraically, we need to consider the area of the hexagonal floor and the cost per square unit of flooring.
The area of a regular hexagon can be calculated using the formula:
Area = (3√3/2) x s²
Where s is the side length of the hexagon.
To find the total cost, we multiply the area by the cost per square unit of flooring (c):
T = c x Area
Substituting the area formula into the equation:
T = c x (3√3/2) x s²
Now, we have expressed the total cost T algebraically in terms of the side length s and the cost per square unit of flooring c.
Thus,
The total cost T of the floor algebraically is T = c x (3√3/2) x s²
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Use the standard normal table to find the specified area.
To the right of z= 1.16
The area to the right of z = 1.16 in the standard normal distribution is approximately 0.1357, or 13.57% as per the given details.
To discover the location to the right of z = 1.16 in the general everyday distribution, we are able to use the same old regular table or a statistical software.
Using the usual normal desk, the values provided are typically for the place to the left of a given z-rating.
However, seeing that the entire region below the same old normal distribution curve is 1, we can discover the place to the right of z = 1.16 by way of subtracting the region to the left of z = 1.Sixteen from 1.
Area to the right = 1 - Area to the left
= 1 - 0.8643
= 0.1357
Thus, the answer is 0.1357.
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Linear Programming Project
You are the new owner of a music shop in Greenwood. The previous owner
fled the city to join the circus as a magician. Your first duty as new owner
and store manager is to create an advertising plan based on the budget
available. You must figure out how many magazines and TV ads to
purchase.
TV ads cost $600 per airing.
Magazine ads cost $1200 per issue.
Your total advertising budget is $9,000.
1. If we let x = TV ads and y = magazine ads, write an inequality for our
advertising budget.
500 x + 1200y <= 9000
2. Due to space limitations, the magazine publishers tell us that we are only
allowed to purchase up to 6 magazine ads. Write an inequality for this
constraint. y<= 6
3. The television station called to say that we are only allowed to purchase
up to 7 TV ads. Write an inequality for this constraint.
X<=7
4. It is impossible to buy a negative number of TV ads. Write an inequality
for this constraint.
-1
-5. It is impossible to buy a negative number of magazine ads. Write an
inequality for this constraint.
-1
Use the points (1, 150) and (6, 900) to estimate the line of best fit.
1. Find the slope of the line.
2. Write the equation of the line
3. What does the slope tell you about how each TV ad affects sale
For every TV ad, CD sales increased by about
Use the points (1, 100) and (8, 800) to estimate the line of best fit.
1. Find the slope of the line.
2. Write the equation of the line.
3. What does the slope tell you about how each magazine ad affer
sales?
CD sales increased by about...?
1)
When the points (1, 150) and (6, 900) are used:
a)
Slope of line is 150.
Given points,
(1, 150) and (6, 900)
Slope of a line passing from two points:
Slope = y2 - y1 / x2 - x1
Slope = 900 - 150 / 6 - 1
Slope = 150
b)
Equation of line :
Y = mx + c
m = slope of line
c = y intercept
Here,
Slope =150
Hence the equation of line is:
y = 150x
c)
For every TV ad, CD sales increased by about 150 as the slope of line is + 150 which indicates the increase in sales of CD .
2)
When the points (1, 100) and (8, 800) ae used:
a)
Slope of line is 100.
Given points,
(1, 100) and (8, 800)
Slope of a line passing from two points:
Slope = y2 - y1 / x2 - x1
Slope = 800 - 100 / 8 - 1
Slope = 100
b)
Equation of line :
Y = mx + c
m = slope of line
c = y intercept
Here,
Slope =100
Hence the equation of line is:
y = 100x
c)
For every magazine ad, CD sales increased by about 100, as the slope of line is + 100 which indicates the increase in sales of CD .
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8:2(2+2)how would solve this problem
Answer:
16
Step-by-step explanation:
To solve this problem, you need to follow the order of operations (PEMDAS):
First, you need to solve the parentheses: 2+2 = 4
Next, you need to solve the multiplication: 2 x 4 = 8
Finally, you need to solve the division: 8 ÷ 2 = 16
What is the vertically opposite angle y in the drawing below? Type in numerical answer only
Answer:
∠x
Step-by-step explanation:
Vertical angles are defined as angles opposite each other where two lines cross. In this case, it is given that you are trying to find the opposite angle of y. By the definition of vertical angles, it will mean that it is ∠x.
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Megan has to spinners. It’s been one is divided into six equal parts. Spenard two is divided into four equal parts. If she spends both spinners what is the probability that’s been a one will land on for an spinner to land on Blue?
The probability that spinner one will land on four, and spinner two will land on blue is (1/24) x (2/24) = 1/288, or approximately 0.3%.
Megan has two spinners with one spinner divided into six equal parts, while the second spinner is divided into four equal parts. The probability that spinner one will land on four, and spinner two will land on blue is required.
The fundamental principle of probability states that the probability of an event happening is the number of favorable outcomes to the total number of outcomes.
To find the probability, it is essential to determine the total number of outcomes by multiplying the number of sections on each spinner.
The total number of outcomes = (Number of sections in spinner one) x (Number of sections in spinner two)
Number of sections in spinner one = 6Number of sections in spinner two = 4Total number of outcomes
= 6 x 4 = 24
The number of outcomes that will result in spinner one landing on four is one, and the number of outcomes that will result in spinner two landing on blue is two.
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1) Graph the parametric path using Slope-Direction diagram.
The curve is x = t^2,y = (t - 1)(t^2 - 4), for t, in [-3,3]
2) Find the length of the pay over the given interval.
(2cost - cost2t, 2sint - sin2t), 0 ≤ t ≤ π/2
Please give step-by-step.
To graph the parametric path x = t^2, y = (t - 1)(t^2 - 4) on a Slope-Direction diagram, we need to plot points by evaluating the expressions for different values of t within the given interval [-3, 3].
First, let's calculate the coordinates for several values of t:
For [tex]t = -3: x = 9, y = 0[/tex]
For[tex]t = -2: x = 4, y = 6[/tex]
For [tex]t = -1: x = 1, y = 0[/tex]
For [tex]t = 0: x = 0, y = -4[/tex]
For[tex]t = 1: x = 1, y = 0[/tex]
For[tex]t = 2: x = 4, y = 6[/tex]
For [tex]t = 3: x = 9, y = 0[/tex]
Plotting these points on the Slope-Direction diagram, we can observe the shape of the curve. The path starts at (9, 0), moves downward to (4, 6), reaches the lowest point at (0, -4), and then goes back up to (4, 6) and (9, 0).
To find the length of the curve given by (2cost - cost2t, 2sint - sin2t) over the interval 0 ≤ t ≤ π/2, we can use the arc length formula:
L = ∫√(dx/dt)² + (dy/dt)² dt
First, calculate the derivatives of x and y with respect to t:
dx/dt = -2cost - 2costsin2t
dy/dt = 2cost - 2sintcos2t
Next, square and add these derivatives:
(dx/dt)² + (dy/dt)² = 4cost² + 4costsin2t + 4cost² + 4sint²cos²2t
Simplify the expression:
(dx/dt)² + (dy/dt)² = 8cost² + 4costsin2t + 4sint²cos²2t
Now, integrate the square root of this expression over the given interval 0 ≤ t ≤ π/2 to find the length of the curve:
L = ∫√(8cost² + 4costsin2t + 4sint²cos²2t) dt, from 0 to π/2
Evaluating this integral will provide the final answer for the length of the curve.
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Last year and investor purchased 115 shares of stock A at $90 per share
The difference in overall loss or gain between sell at the current day's high price or low price is found tp be the difference in overall gain as $280.10
The third option is correct.
How do we calculate?For stock A:High price value: 115 shares * $105.19 per share = $12,084.85
Low price value: 115 shares * $103.25 per share = $11,858.75
For stock B:High price value: 30 shares * $145.18 per share = $4,355.40
Low price value: 30 shares * $143.28 per share = $4,298.40
The overall value at high price:
$12,084.85 + $4,355.40
= $16,440.25
The overall value at low price:
$11,858.75 + $4,298.40
= $16,157.15
In conclusion, the difference in overall gain or loss:
$16,440.25 - $16,157.15 = $280.10
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Please help please I’m failing every thing I’m trying to pass
Answer:
[tex]11\frac{7}{10}[/tex]
Step-by-step explanation:
[tex]\displaystyle 3\frac{1}{4}+\biggr(3\frac{1}{4}+5\frac{1}{5}\biggr)\\\\3\frac{5}{20}+3\frac{5}{20}+5\frac{4}{20}\\\\(3+3+5)+\biggr(\frac{5}{20}+\frac{5}{20}+\frac{4}{20}\biggr)\\\\11+\frac{14}{20}\\\\11\frac{7}{10}[/tex]
Again, make sure all denominators are the same by using the least common denominator (in this case it's 20 since 5*4=20).
How can you simplify the exponential expression
Answer:
[tex]( { \frac{ {r}^{2} }{s} })^{3} = \frac{ {r}^{2 \times 3} }{ {s}^{1 \times 3} } = \frac{ {r}^{6} }{ {s}^{3} } [/tex]
14. A nutritionist collected information about different
brands of beef hot dogs. She made a table showing
the number of Calories and the amount of sodium in
each hot dog.
a. Write an equation for the line of best fit.
Round all values to the nearest tenth if
needed.
Calories per
Beef Hot Dog
186
181
176
149
184
190
158
139
b. Write the correlation coefficient for the line of best fit.
Round to the nearest hundredth.
Milligrams of Sodium
per Beef Hot Dog
495
477
425
322
482
587
370
322
c. What does the correlation coefficient tell you about your line of best fit?
a. The equation of the line of best fit is y = 1.2x + 375.
b. The correlation coefficient for the line of best fit is 0.97.
c. The correlation coefficient of 0.97 is very close to 1, which indicates that there is a strong positive correlation between the number of calories and the amount of sodium in beef hot dogs.
How to explain the informationMean of x-values: (186 + 181 + 176 + 149 + 184 + 190 + 158 + 139) / 8 = 169
Mean of y-values: (495 + 477 + 425 + 322 + 482 + 587 + 370 + 322) / 8 = 437.5
Calculate the slope of the line of best fit.
Slope = (y2 - y1)/(x2 - x1) = (587 - 495)/(190 - 186) = 1.2
Calculate the y-intercept of the line of best fit.
y-intercept = mean of y-values - slope * mean of x-values
= 437.5 - 1.2 * 169
= 375
The equation of the line of best fit is y = 1.2x + 375.
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5x = 50. Find x. PLS HELP PLS
Answer:
[tex]\Huge \boxed{\boxed{x = 10}}[/tex]
Step-by-step explanation:
To solve the equation [tex]5x = 50[/tex] and find the value of [tex]x[/tex], we need to isolate [tex]x[/tex] on one side of the equation.
Step 1: Divide both sides of the equation by 5.
[tex]\frac{5x}{5} = \frac{50}{5}[/tex]
Step 2: Simplify this expression
[tex]x = 10[/tex]
Therefore, the solution to the equation [tex]\boldsf{5x = 50}[/tex] is [tex]x = 10[/tex].
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How many inches are in 5 1/4 feet enter only a number
Answer:
64 inches.
Step-by-step explanation:
To find the amount of inches in 5 1/4 feet, you must first multiply 12 by 5 because there are 12 inches in one foot.
12x5=60.
The next step in order to complete the equation would be to find how many inches are in 1/4 of a foot.
To find how many inches are in 1/4 of a foot you must divide 12 by 4.
12 divided by 4= 4.
Now, you must as the two numbers together to get the total number of inches.
60+4=64.
This, 64 is your final answer.
Hope this helps! :)
Could u help me to pick the answer please I really need answer right now
Answer:
A
Step-by-step explanation:
One of your customers is delinquent on his accounts payable balance. You’ve mutually agreed to a repayment schedule of $570 per month. You will charge .97 percent per month interest on the overdue balance. If the current balance is $14,790, how long will it take for the account to be paid off?
Note: Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.
Answer:
To calculate the time it will take to pay off the account, we can use the formula for the future value of an annuity:
PV = PMT * ((1 - (1 + r)^(-n)) / r)
Where:
PV = Present value (current balance)
PMT = Payment amount per period
r = Interest rate per period
n = Number of periods
In this case:
PV = $14,790
PMT = $570
r = 0.0097 (0.97% expressed as a decimal)
n = ?
Plugging in the values, we can solve for n:
14,790 = 570 * ((1 - (1 + 0.0097)^(-n)) / 0.0097)
Let's solve this equation to find the value of n:
((1 + 0.0097)^(-n)) = 1 - (14,790 * 0.0097) / 570
((1 + 0.0097)^(-n)) = 0.9742105
Taking the logarithm of both sides:
-n * log(1.0097) = log(0.9742105)
n = log(0.9742105) / log(1.0097)
Using a calculator, we find that n is approximately 28.56.
Therefore, it will take approximately 28.56 months (or 28 months and 17 days) to pay off the account.
Step-by-step explanation:
Find the perimeter of the figure. All angles in the figure are right angles.
A. 82 cm
B. 59 cm
C. 77 cm
D. 356.25 cm
Answer:
A. 82 cm
Step-by-step explanation:
You want the perimeter of the L-shaped figure that is 12.5 cm high and 28.5 cm wide.
Side lengthsThe top two horizontal line segments have the same total length as the bottom horizontal line segment. We don't need to figure what the missing length is, because we only need their total for the perimeter.
The right side two vertical line segments have the same total length as the left vertical line segment. As with the horizontal segments, we don't need to figure out the missing length, because we only need their total for the perimeter.
PerimeterThe perimeter is the sum of horizontal line segments plus the sum of vertical line segments:
2(28.5 cm) + 2(12.5 cm) = 57 cm + 25 cm = 82 cm
The perimeter of the figure is 82 cm.
<95141404393>
The Puyer Corporation makes and sells only one product called a Deb. The company is in the process of prepon
The following budget data are available
Advertising
Executive salaries
Depreciation on office equipment
Other
$ 51,960
$ 21,900
Variable Cost Per
Deb Sold
All of these expenses (except depreciation) are paid in cash in the month they are incurred
If the company has budgeted to sell 16,900 Debs in February, then the total budgeted fixed selling and administra
The Puyer Corporation has budgeted fixed selling and administrative expenses of $73,860 for February.
How to solveThis includes advertising of $51,960, executive salaries of $21,900, and other expenses of $9,900.
The variable cost per Deb sold is $2.50. If the company sells 16,900 Debs in February, then the total budgeted selling and administrative expenses will be $89,760.
The total budget spending is given as:
$73,860 + (16,900 * $2.50) = $89,760
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A polynomial f(x) has a lead coefficient of one and exactly three distinct zeros. Find the polynomial that uld go with this (multiply it all out) x = -2 is a zero with a multiplicity of one is a zero with a multiplicity of two x = 3 X = 1 is a zero with a multiplicity of one 0
If x = -2 is a zero with a multiplicity of one, x = 3 is a zero with a multiplicity of two, and x = 1 is a zero with a multiplicity of one, then the polynomial can be written in factored form as:
[tex]f(x) = (x + 2)(x - 3)^2(x - 1)[/tex]
To find the polynomial in expanded form, we can use the distributive property and the rules of exponents:
[tex]f(x) = (x + 2)(x - 3)(x - 3)(x - 1)\\= (x^2 - x - 6)(x - 3)(x - 1)\\= (x^3 - 4x^2 + 3x + 18)(x - 1)\\= x^4 - 5x^3 + 6x^2 + 4x + 18[/tex]
Therefore, the polynomial that has a lead coefficient of one and exactly three distinct zeros, with x = -2 as a zero with multiplicity of one, x = 3 as a zero with multiplicity of two, and x = 1 as a zero with multiplicity of one, is:
[tex]f(x) = x^4 - 5x^3 + 6x^2 + 4x + 18[/tex]
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if the AREA of a rectangular garden is x^2-36 and the length is x^2-2x-24, find an expression to represent the width of the garden.
To find an expression representing the width of the rectangular garden, we need to use the given information about the area and length of the garden.
The formula for the area of a rectangle is:
Area = Length × Width
We are given that the area of the garden is x^2 - 36, and the length is x^2 - 2x - 24.
Let's substitute these values into the formula:
x^2 - 36 = (x^2 - 2x - 24) × Width
To isolate the width, we divide both sides of the equation by (x^2 - 2x - 24):
Width = (x^2 - 36) / (x^2 - 2x - 24)
Therefore, an expression representing the width of the garden is:
Width = (x^2 - 36) / (x^2 - 2x - 24)
abc by cd
Help!!!!!!!
Hello!
abc * cd
= a * b * c * c * d
= abc²d
[tex] \bold{abc \: \: by \: \: cd}[/tex]
Step-by-step explanation :[tex] \bold{(abc) \cdot(cd)}[/tex]
[tex] \sf{abc {}^{1 + 1} d}[/tex]
[tex] \sf{abc {}^{2} d}[/tex]
The general rule for multiplication of monomials says that: multiply the coefficients and then write the letters of the factors in alphabetical order. Each letter is given an exponent equal to the sum of the exponents it has in the factors. The sign of the product result will be given by the Law of Signs.
Form a polynomial f(x) with real coefficients having the given degree and zeros.
Degree 5: zeroes:3, -i;9+i
Let a represent the leading coefficient. The polynomial is f(x)=a(
The polynomial f(x) with real coefficients and the given zeros is:
f(x) = x^3 - (12 + i)x^2 + (x - 3 - 2i)x + 27x + (27 + 3i)
To form a polynomial with degree 5 and the given zeros, we can start by writing the factors corresponding to each zero.
The zero 3 gives us the factor (x - 3).
The zero -i gives us the factor (x + i) since complex zeros always come in conjugate pairs.
The zero 9+i gives us the factor (x - (9+i)).
Now, we can multiply these factors together to obtain the polynomial:
f(x) = (x - 3)(x + i)(x - (9+i))
Next, we simplify the expression:
f(x) = (x - 3)(x + i)(x - 9 - i)
Expanding the product, we have:
f(x) = (x^2 + xi - 3x - 3i)(x - 9 - i)
Multiplying further:
f(x) = (x^3 - 9x^2 - ix^2 + xi - 3x^2 + 27x + 3ix - 3xi - 27i - 3x + 27 + 3i)
Combining like terms:
f(x) = x^3 - (9 + i)x^2 - 3x^2 + (x - 3 - 3i)x + 27x + (27 + 3i)
Simplifying:
f(x) = x^3 - (12 + i)x^2 + (x - 3 - 2i)x + 27x + (27 + 3i)
The polynomial f(x) with real coefficients and the given zeros is:
f(x) = x^3 - (12 + i)x^2 + (x - 3 - 2i)x + 27x + (27 + 3i)
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A man undertakes to pay off a debt of Rs. 622500 by monthly installments, he pays Rs. 10000 in the first month and continually increases the installments in every subsequent month by Rs. 100. In what time will the debt be cleared up?
The debt will be cleared in approximately 158 months.
In how many months will the debt be cleared?The monthly installment starts at Rs. 10,000 and increases by Rs. 100 each month. We can set up an arithmetic progression to represent the installment amounts:
10,000, 10,100, 10,200, ...
The nth term of this arithmetic progression can be calculated using the formula:
an = a1 + (n-1)d
where:
an = nth term of the sequence
a1 = first term of the sequence
d = common difference between the terms
In this case, a1 = 10,000 and d = 100.
Now we need to find the value of n such that the sum of the first n terms of the sequence exceeds or equals Rs. 622,500.
The sum of the first n terms of an arithmetic sequence can be calculated using the formula:
Sn = (n/2)(2a1 + (n-1)d)
We need to solve the equation:
Sn >= 622,500
Substituting the values, we have:
(n/2)(2*10,000 + (n-1)*100) >= 622,500
Simplifying the equation, we can solve for n:
n^2 + 199n - 614000 = 0
Using the quadratic formula, we find that:
n ≈ 157.37 or n ≈ -356.37
Since we cannot have a negative number of months, we take n ≈ 157.37. Therefore, it will take approximately 158 months (rounded up) to clear the debt.
Hence, the debt will be cleared in approximately 158 months.
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