Commercial paper with a maturity of 270 number of days or less is a type of short-term debt instrument that can be used for a variety of purposes, including funding operating expenses and managing cash flow.
It is typically issued by large corporations with high credit ratings. Commercial paper typically has a lower interest rate than other types of debt instruments, making it an attractive option for businesses seeking to raise capital quickly.
Commercial paper is a type of short-term debt instrument with a maturity of 270 number of days or less. It is typically issued by large corporations with high credit ratings and has a lower interest rate than other types of debt instruments, making it an attractive option for businesses seeking to raise capital quickly.
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When a customer buys a family-sized meal at a certain restaurant, they get to choose 3 side dishes from 9 options. Suppose a customer is going to choose 3 different side dishes.
How many groups of 3 different side dishes are possible?
There amount are 504 different groups of 3 side dishes that a customer can choose from.
There are 9 possible side dishes, and the customer can choose any 3 of them. The number of different groups of 3 side dishes can be determined using the formula:
nCr = n!/(r!(n-r)!).
In this case,
n = 9 and r = 3,
so the number of possible groups is
9!/(3!(9-3)!) = 504.
When a customer buys a family-sized meal at a certain restaurant, they get to choose 3 side dishes from 9 options. The number of different groups of 3 side dishes possible for the customer to choose from can be determined using the formula for combination: nCr = n!/(r!(n-r)!). In this case, n = 9 (the number of possible side dishes) and r = 3 (the number of side dishes chosen). Applying the formula, the number of possible groups of 3 side dishes is 9!/(3!(9-3)!) = 504. This means that there are 504 different groups of 3 side dishes that a customer can choose from.
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Write an equation in point-slope form that passes through the points (2, 4) and (-3, -6).
The point-slope form of the equation that passes through the points [tex](2, 4)[/tex]and [tex](-3, -6)[/tex] is: [tex]y - 4 = -5/5 (x - 2)[/tex].
To find the equation in point-slope form, you can use the point-slope formula: [tex]y - y1 = m(x - x1)[/tex]
Where [tex](x1, y1)[/tex] is a point on the line, and m is the slope of the line, which can be found by using the two points given:
[tex]m = (y2 - y1) / (x2 - x1)[/tex]
[tex]m = (-6 - 4) / (-3 - 2) = -10/5 = -2[/tex]
Then you can substitute the values into the point-slope formula:
[tex]y - 4 = -2(x - 2)[/tex]
[tex]y = -2x + 8[/tex]
[tex]y - 4 = -5/5 (x - 2)[/tex]
This is the final equation in point-slope form that passes through the points [tex](2, 4)[/tex] and [tex](-3, -6).[/tex]
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Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle.
The area of the region enclosed by the given curves is 32 units2.
The region enclosed by the given curves is bounded by the equation y = x2 and y = 4. The region is a parabolic region and needs to be integrated with respect to y. A typical approximating rectangle for this region is shown in the diagram below.
The area of this region can be expressed as the integral:
A = ∫y=x2→4 y dx
This integral can be evaluated by splitting the integral into two parts: A = ∫y=x2→4 x2 dx + ∫y=x2→4 4dx
After evaluating the integrals, the area of the region is:
A = (1/3)x3 + 4x |y=x2→4
A = (1/3)(42) + 4(4 - 22)
A = 32
Therefore, the area of the region enclosed by the given curves is 32 units2.
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find the indicated probability round to three decimal places
The probability of randomly selecting a blue ball is 3/14, or 0.214 rounded to three decimal places.
Formula: P(blue ball) = n(blue ball)/n(total balls)
P(blue ball) = 3/14 = 0.214
The probability of randomly selecting a blue ball is 0.214. To calculate this probability, the formula P(blue ball) = n(blue ball)/n(total balls) was used. This formula is used to find the probability of an event occurring, in this case randomly selecting a blue ball. The numerator of the equation is the number of blue balls in the bag, which is 3, while the denominator is the total number of balls in the bag, which is 14. Therefore, the probability of randomly selecting a blue ball is 3/14, or 0.214 rounded to three decimal places.
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Complete question:A bag contains 6 yellow, 3 blue, and 5 red balls. Find the probability of randomly selecting a blue ball.
public transportation and the automobile are two methods an employee can use to get to work each day. samples of travel times recorded for each method are shown. times are in minutes.
a. The sample mean time to get to work for public transportation is 31.3 minutes and the sample mean time to get to work for the automobile is 32.1 minutes.
b. The sample standard deviation for public transportation is 4.2 minutes and the sample standard deviation for the automobile is 1.7 minutes.
c. Based on the results from parts (a) and (b), public transportation should be preferred. The sample mean time for public transportation is slightly lower than the sample mean time for the automobile, and the sample standard deviation for public transportation is much lower than the sample standard deviation for the automobile.
This indicates that public transportation is more consistent and reliable than the automobile, and therefore should be preferred as a method of transportation to get to work each day.
Complete question:
Public transportation and the automobile are two methods an employee can use to get to work.
each day. Samples of times recorded for each method are shown. Times are in minutes.
Public Transportation: 28 29 32 37 37 33 25 29 32 41 34
Automobile: 29 31 33 32 34 30 31 32 35 33
a. Compute the sample mean time to get to work for each method.
b. Compute the sample standard deviation for each method.
c. On the basis of your results from parts (a) and (b), which method of transportation should be preferred? Explain.
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Please help! Thank you so much!
What is the exponential function?
An exponential function is a type of mathematical function in which the output (the y-value) is a constant multiplied by a fixed number (the base) raised to the power of the input (the x-value). The general form of an exponential function is y = ab^x, where a and b are constants and x is the input value.
1 .To estimate f(1/3) using the graph points (0, 12) and (1, 0.75), we can use the fact that an exponential function has the form y = ab^x, where a and b are constants and x is the input value. By looking at the graph, we can see that the y-intercept (when x = 0) is 12, which means that a = 12. We can also see that the function passes through the point (1, 0.75), which means that f(1) = 0.75. Using this information, we can write the equation for the function as:
y = 12b^x
f(1/3) = 12b^(1/3)
Since we don't know the value of b, it's impossible to know the exact value of f(1/3) from the given information. However, it tells us that the amount of medicine in the bloodstream decreases exponentially and how fast it decreases depends on the value of b.
2. To find the equation that defines f, we can use the point (1, 0.75). We know that f(1) = 0.75, so we can substitute this into the equation we found earlier:
0.75 = 12b^1
0.75 = 12b
b = 0.0625
So, the equation that defines f is:
y = 12 * 0.0625^x
or
y = 0.75 * 0.0625^x
This equation tells us that the amount of medicine in the bloodstream decreases exponentially with time, with a rate determined by the value of b = 0.0625.
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The probability of a randomly selected adult having a rare disease for which a diagnostic test has been developed is 0.001. The diagnostic test is not perfect. The probability the test will be positive (indicating that the person has the disease) is 0.99 for a person with the disease and 0.02 for a person without the disease. The proportion of adults for which the test would be positive isA) 0.00099B) 0.01998C) 0.02097D) 0.02100
Adults make up 0.02100 of the probability for whom the test would be positive.
The proportion of adults for which the test would be positive can be calculated by multiplying the probability of a randomly selected adult having the rare disease (0.001) by the probability that the test will be positive for a person with the disease (0.99) and adding it to the probability that the test will be positive for a person without the disease (0.02).
0.001 x 0.99 = 0.00099
0.00099 + 0.02 = 0.02099
0.02099 = 0.02100 (rounded)
Therefore, the proportion of adults for which the test would be positive is 0.02100.The proportion of adults for which the test would be positive is 0.02100, which is calculated by taking into account the probability of a randomly selected adult having the rare disease (0.001) and the probability that the test will be positive for a person with the disease (0.99) and a person without the disease (0.02). This reflects how the test is not perfect and can sometimes give false positives or false negatives. The result of 0.02100 indicates that out of every 1000 adults, 2 of them will have a positive test result.
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DUE SOON!
PLEASE HELP EXPLAIN THIS IM SO CONFUSED!
Answer:
$20.44
Step-by-step explanation:
The average of a set of numbers is defined as:
[tex]\dfrac{\textrm{sum of values}}{\textrm{number of values}}[/tex]
In this problem, we are shown a table where each row has a coupon value in the left column and the number of that coupon value in the right column (e.g., if we look at the top row, we can see there are 70 coupons each valued at $10).
So, the sum of values in this problem (i.e., the total number of dollars given out by the store in the form of coupons) is defined as the sum of the product of each row.
[tex]\textrm{sum of values} = (\$10 \times 70) + (\$20 \times 40) + (\$40 \times 20) + (\$60 \times 4) + (\$120 \times 2)[/tex]
[tex]\textrm{sum of values} = \$700 + \$800 + \$800 + \$240 + \$240[/tex]
[tex]\textrm{sum of values} = \$2780[/tex]
The number of values in this problem is just the sum of the numbers in the right column (i.e., the number of coupons given out).
[tex]\textrm{number of values} = 70 + 40 + 20 + 4 + 2[/tex]
[tex]\textrm{number of values} = 136[/tex]
Finally, to answer the problem, we can plug the two numbers that we just solved for into the formula for the average of a set.
[tex]\textrm{average savings} = \dfrac{\textrm{value of all tickets}}{\textrm{number of tickets}}[/tex]
[tex]\textrm{average savings} = \dfrac{\$2780}{136}[/tex]
[tex]\textrm{average savings} \approx \$20.44[/tex]
What number should be added to −96 to get a sum of 9 ?
The number 105 should be added to - 96 to get a sum of 9.
What is Addition?The process of combining two or more numbers is called the Addition. The 4 main properties of addition are commutative, associative, distributive, and additive identity.
Now,
Let the value of number = x
So, We can formulate;
⇒ x + (-96) = 9
Solve for x;
⇒ x - 96 = 9
Add 96 both side,
⇒ x - 96 + 96 = 9 + 96
⇒ x = 105
Thus, The number 105 should be added to - 96 to get a sum of 9.
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What is the value of X
Answer:
A) 540
B) x = 115/3
Step-by-step explanation:
A) to find the sum of the interior angles of any gon, use this formula: 180(n-2); n = the number of sides of the gon you are given (ex. octagon will be n=8). Therefore, a pentagon will be: 180(5-2) = 540
B) To find x, we can add all the angles and make them equal to 540.
(3x+50) + (2x+60) + (3x-30) + 2x + 2x = 540
12x = 460
x = 115/3
C) Plug-in x for every angle
The population parameters below describe the full-time equivalent number of students (FTES) each year at Lake Tahoe Community College from 1976–1977 through 2004–2005. μ = 1000 FTES median = 1,014 FTES σ = 474 FTES first quartile = 528.5 FTES third quartile = 1,447.5 FTES n = 29 years
a. 75% of all years have an FTES at or below ___.
1447.5 is 75% of all years have an FTES at or below .
What does FTEs mean?
An employee's scheduled hours are divided by the employer's hours for a full-time workweek to determine their full-time equivalent (FTE). Employees who are scheduled to work 40 hours per week for an employer are considered 1.0 FTEs.
A worker's scheduled hours are divided by the company's work hours on a weekly full-time basis to determine their full-time equivalent (FTE). A 40-hour workweek means that there will be 1.0 FTEs of employees working for that company throughout that time.
We know that,
75% of all the data are at or below the value of third quartile.
Hence, required correct answer is,
75% of all years have an FTES at or below 1447.5 .
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Let (-3, -5) be a point on the terminal side of an angle in standard position. find the exact values of the six trigonometric functions of the angle.
The value of the six trigonometric functions as ( -3 ,-5 ) is the point on the terminal side of an angle is given by:
sin α = -5 /√34 cosec α = - √34 / 5
cos α = -3 / √34 sec α = -√34 / 3
tan α = 5 / 3 cot α = 3 /5
As given in the question,
In standard position ( -3 ,-5 ) represents the terminal side of an angle.
Here in right angled triangle,
Base = -3
Height = -5
Using Pythagoras theorem ,
Hypotenuse = √ ( -3 )² + ( -5 )²
= √9 + 25
= √34
Value of the six trigonometric functions are given by :
sin α = -5 /√34
cos α = -3 / √34
tan α = 5 / 3
cosec α = - √34 / 5
sec α = -√34 / 3
cot α = 3 /5
Therefore, the value of the six trigonometric functions as per the given terminal point of an angle is equal to :
sin α = -5 /√34 cosec α = - √34 / 5
cos α = -3 / √34 sec α = -√34 / 3
tan α = 5 / 3 cot α = 3 /5
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Vincent makes handcrafted dining tables, and he is trying to decide how many tables to produce. He can sell each dining table for $1,000. The cost of the first table is $900, for the second it's $1,100. For each additional table he produces, the marginal cost of each table increases by $200. How many dining tables should Vincent produce, and what is the total cost of his production?
The total cost of production of six dining tables is $1300.
How many tables to be produced?Production is the act of combining a variety of material and immaterial inputs to create an output. In a perfect world, this output would be a good or valuable good or service for people to use. The three main production factors of land, labor, and capital are known as primary producers of goods or services.
These crucial inputs are not significantly altered by the output process or the end product, nor are they made a crucial component of it. In classical economics, materials and energy are regarded as secondary factors because they are byproducts of land, labor, and capital. Technology and entrepreneurship are sometimes seen as evolved variables in production in addition to the conventional components of production.
Each dining table for $1,000
The second it's $1,100.
The marginal cost of each table increases by $200
The total cost of his production is
Fixed cost + Variable cost
= $1000 + 300= $1300
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P(x) =2x³ + 2x² - 3x - 3 is divisible by:
A. x - 1
B. x + 1
C. x + 2
D. x + 3
E. x - 2
Answer:
Option B
Step-by-step explanation:
If you replace the X in the P(X) it becomes -1 and then you'll change every x in the equation to -1
2(-1)³ + 2(-1)² -3(-1) -3
which = -2 +2 +3 -3
= 0
Since the answer is 0 it means x + 1 is a factor of 2x³ + 2x² - 3x -3
Answer:
B. x + 1-----------------------
Factorize the given polynomial:
P(x) =2x³ + 2x² - 3x - 3 = 2x²(x + 1) - 3(x + 1) =(x + 1)(2x² - 3)As we see the polynomial is divisible by x + 1.
Correct choice is B.
Does anybody know this?Ice at it for an hour
Answer: 3x + 5
x = x + 2
3 (x+2) -1
3x + 6 - 1
3x+5
Convert 25 dollars to dimes
Answer:
Step-by-step explanation:
250
Answer:250
Step-by-step explanation: just add 10 till you get 250
Amelia selects a counter at random 400 times and records the number of times the counter is black, yellow, green, purple or red.
Black - 40 Yellow- 100 Green- 100 Purple- 80 red- 80
Work out the relative frequency of landing on yellow.
The relative frequency of yellow is 25%, which means that yellow was the outcome of 25 out of every 100 selections.
What is the relative frequency?
Relative frequency is the ratio of the number of times an event occurs to the total number of data points.
The relative frequency of landing on yellow is 100/400 = 0.25 or 25%.
This is found by dividing the number of times yellow is landed on (100) by the total number of trials (400).
This means that out of the 400 times that Amelia selected a counter, 25% of those times the counter was yellow.
It is a measure of how often a particular outcome occurs in comparison to the total number of trials or selections.
Hence, the relative frequency of yellow is 25%, which means that yellow was the outcome of 25 out of every 100 selections.
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Calculate s20 for the arithmetic sequence in which a=32 and the common difference is d=2.9
It is possible to find the value of S20 by applying arithmetic progression. similar by applying the sum formula.
Formula is used to determine S20.
First term (a) here equals 32, and common difference (d) is equal to 2.9
Sn = n/2 × ( 2a + (n-1) × d)
S20 = 20/2 × ( 2×32 + ( 20 - 1 ) × 2.9 )
S20 = 20/2 × ( 64 + (19 × 2.9) )
S20 = 20/2 × ( 64 + 55.1 )
S20 = 20/2 × 119.1
S20 = 1191
Therefore, value of S20 is 1191
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Answer:
Step-by-step explanation:
[tex]s_{n}=\frac{n}{2} [2a+(n-1)d]\\s_{20}=\frac{20}{2} [2 \times 32+(20-1)\times 2.9]\\=10[64+19\times 2.9]\\=640+551\\=1191[/tex]
g a cusp point (or a point where the curve changes direction abruptly instead of smoothly) can occur when:
Countless pointy corners are called cusps. A cusp is a place where a vertical tangent exists, but where one side's derivative is positive and the other side's derivative is negative.
The above example of a paradigm is y=x23. The derivative's upper limit as you move left toward zero is. As you move toward zero from the right, the derivative's limit is +. A vertical tangent exists at x=0.
Cusps have vertical tangents (when describing functions). A vertical tangent, however, is not a cusp by itself. The y=x13 curve is smooth, showing no behavior like a pointed needle, although it does have a vertical tangent at the origin.
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In Exercises 1-8, find the area of the shaded region. The radius of each circle is r. If two circles are shown, r is the radius of the smaller circle and R is the radius of the larger circle. 1. r= 6 cm 2. r= 8 cm 3. r = 16 cm 160° 24001 4. r= 2 cm 5. r= 8 cm 6. R= 7 cm r= 4 cm
The areas of the shaded region of the given polygons is as follows
1. Area = 18.84cm²
2. Area = 67cm²
3. Area = 602.88cm²
4. Area = 1.14cm²
5. Area = 182.72cm²
6. Area = 33 cm²
What is the area of the shaded region?The area of the shaded area is the difference between the total area of the polygon and the area of the portion of the polygon that is not shaded. In polygons, the area of the shaded component might appear in two different ways. A polygon's sides or its center are both potential locations for the shaded area.
Here, we have
1. r= 6 cm, θ = 60°
Area of shaded region = πr² × θ/360
Area = 3.14 × 6× 6 × 60/360
Area = 18.84cm²
2. r= 8 cm
θ = 360-240 = 120
Area = 3.14 × 8× 8 × 120/360
Area = 67cm²
3. r = 16 cm
θ = 360 - 90 = 270
Area = 3.14 × 16× 16 × 270/360
Area = 602.88cm²
4. r= 2 cm
θ = 90
Area = 3.14 × 2× 2 × 90/360
Area = 1.14cm²
5. r= 8 cm
θ = 270
Area = πr² × θ/360 + 1/2(bh)
Area = 3.14 × 8× 8 × 270/360 + 1/2(8×8)
Area = 182.72cm²
6. R = 7 cm
r = 4 cm
Area = πR² - πr²
= 22/7 × 7² - 22/7 × 4²
= 49 - 16
= 33 cm²
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Full question:
We want to know how many people plan on buying a new car this year. Determine the sample size needed for a one proportion confidence interval to estimate this with a margin of error of 0.06 and a 95% confidence interval. We don't have any idea what the population proportion would be.
The required sample size for the confidence interval is given as follows:
n = 267.
What is a confidence interval of proportions?A confidence interval of proportions has the bounds given by the rule presented as follows:
[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
In which the variables used to calculated these bounds are listed as follows:
[tex]\pi[/tex] is the sample proportion, which is also the estimate of the parameter.z is the critical value.n is the sample size.The margin of error is defined as follows:
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
The confidence level is of 95%, hence the critical value z is the value of Z that has a p-value of [tex]\frac{1+0.95}{2} = 0.975[/tex], so the critical value is z = 1.96.
The parameters for this problem are given as follows:
[tex]M = 0.06, \pi = 0.5[/tex]
The estimate is of 0.5 as it is when the highest sample size is needed, and the sample size is obtained as follows:
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
[tex]0.06 = 1.96\sqrt{\frac{0.5(0.5)}{n}}[/tex]
[tex]0.06\sqrt{n} = 1.96 \times 0.5[/tex]
[tex]\sqrt{n} = \frac{1.96 \times 0.5}{0.06}[/tex]
[tex](\sqrt{n})^2 = \left(\frac{1.96 \times 0.5}{0.06}\right)[/tex]
n = 267.
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Omg please answer this!! I really need help. Evaluate (8+t)^3 - 6 when t =2.
Answer: 994
Step-by-step explanation:
(8+(2))^3-6= (10)^3-6= 1000-6= 994
Answer:
The value of the given expression is 994.
The given expression is (8 + t)³ - 6.
Step-by-step explanation:
Substitute t=2 in the given expression and simplify. That is,
(8 + t)³ - 6
= (8 + 2)³ - 6
= 10³ - 6
= 1000-6
=994
Therefore, the value of the given expression is 994.
Consider an election with 521 votes
If there are 8 candidates, what is the smallest number of first-place votes a candidate could win with under the Plurality method?
Prove that there exists a positive integer n < 10^6 such that 5^n has six consecutive zeros in its decimal representation
It has been proven that there exists a positive integer n < 10^6 such that 5^n has six consecutive zeros in its decimal representation. This is done by induction, where the base case is when n = 6, and the inductive step is when n is increased by 1.
We can prove this by induction.
When n = 6, 5^6 = 15,625 and it has six consecutive zeros in its decimal representation.
Suppose that there exists a positive integer n < 10^6 such that 5^n has six consecutive zeros in its decimal representation.
Let k = n + 1. Then 5^k = 5^n * 5 and it has six consecutive zeros in its decimal representation.
Thus, by induction, we can conclude that there exists a positive integer n < 10^6 such that 5^n has six consecutive zeros in its decimal representation.
It has been proven that there exists a positive integer n < 10^6 such that 5^n has six consecutive zeros in its decimal representation. This is done by induction, where the base case is when n = 6, and the inductive step is when n is increased by 1.
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How many oz. does the bag of apples weigh?
Answer:
68 oz.
Step-by-step explanation:
The bag weighs 4.25lbs
1lb=16oz.
4.25*16=68oz.
Suppose the cost of a business property is $7,100,000 and a company depreciates it with the straight-line method. If V is the value of the property after x years and the line representing the value as a function of years passes through the points (90,1520000) and (100,900000), write the equation that gives the annual value of the property.
Answer: [tex]V=-62000x+15200000[/tex]
Step-by-step explanation:
The slope of the line is [tex]\frac{900000-1520000}{100-90}=-62000[/tex].
Using point-slope form, the equation is:
[tex]V-900000=-62000(x-100)\\\\V-900000=-62000x+620000\\\\V=-62000x+15200000[/tex]
a line that has a slope or 1/5 and passes through (-10,4)
Answer: [tex]y=\frac{1}{5} x+6[/tex]
Step-by-step explanation:
(-10,4) **substitute these points into the equation below**
y = [tex]\frac{1}{5} x+c[/tex]
4 = [tex]\frac{1}{5}[/tex][tex](-10)[/tex][tex]+c[/tex] (make c the subject)
c = 4 + 2
c = 6 (this is the y-intercept that is needed when creating the equation of the line)
therefore: [tex]y=\frac{1}{5} x+6[/tex]
[tex]\sf y =\dfrac{1}{5} x+6.[/tex]
Step-by-step explanation:1. Identify the data.We're given the slope and an ordered pair the function passes through.
[tex]\sf Slope (m)=\dfrac{1}{5}[/tex]
[tex]\sf Point: (-10,4)\\ \\Therefore:\\x_{1}=-10\\y_{1} =4[/tex]
2. Use the formula for calculating linear equations based on the slope and a point.Here's that formula: [tex]\sf y-y_{1} =m(x-x_{1} )\\ \\[/tex]
3. Substitute the variables in the formula by the identified values on step 1.[tex]\sf y-(4) =(\dfrac{1}{5} )(x-(-10) )\\ \\[/tex]
4. Calculate.[tex]\sf y-(4) =(\dfrac{1}{5} )(x+10 )\\ \\\\\sf y-(4) =(\dfrac{1}{5} )(x)+(\dfrac{1}{5} )(10)\\ \\ \\\sf y-(4) =\dfrac{1}{5} x+(\dfrac{10}{5} )\\ \\ \\\sf y-4 =\dfrac{1}{5} x+2\\ \\ \\\sf y-4+4 =\dfrac{1}{5} x+2+4\\ \\ \\\sf y =\dfrac{1}{5} x+6[/tex]
5. Verify the answer.From plain sight we can tell that the slope of this equation is indeed 1/5, because the value that multiplies "x" is 1/5 when the equation is solved for "y". Now, to make sure it passed through the given point, substitute "x" by "-10" and make sure it returns a value of "4" for "y".
[tex]\sf y =\dfrac{1}{5} (-10)+6\\ \\ \\\sf y =\dfrac{-10}{5}+6\\ \\ \\\sf y =-2+6\\ \\ \\y=4[/tex]
The answer is correct!
Therefore, the equation of the line that has a slope or 1/5 and passes through (-10,4) is: [tex]\sf y =\dfrac{1}{5} x+6[/tex].
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for which of the following functions would the quotient rule be considered the best method for finding the derivative?
h(x)=f(x)/g(x) separating a fraction like this, the quotient rule is applicable.
The numerator is the first function.
The denominator is the second function.
In essence, it is [(derivative of first function)* second function - (derivative of second function)* first function] divided by the square of the second function. So the functions would the quotient rule be considered the best method for finding the derivative-
When separating a fraction like this, the quotient rule is applicable. f(x)/g(x)
This is differentiated by squaring the denominator and using the product rule to the numerator:
h(x)=f(x)/g(x)
therefore:
h′(x)=(f′(x)g(x)−f(x)g′(x))/g2(x)
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Suppose b, c R. Define T: P(R) → R2 by Tp=(3p(4) + 5p'(6)+bp(1)p(2), x3 p(x) dx + c sin p(0) Show that T is linear if and only if b = c = 0.
Therefore, additivity and homogeneity of T are satisfied, T is linear.
T must be linear for both b and c to be true. T is linear, hence additivity is true for every p, q, and P. (R). In order to make our computations as straightforward as feasible, it would be a good idea for us to use straightforward polynomials in P(R). p, q ∈ P(R), where p(x) = [tex]\frac{\pi }{2}[/tex] and q(x) = [tex]\frac{\pi }{2}[/tex] for all x ∈ R and so we have
T(p + q) =(3(p + q)(4) +5(p + q)'(6)+b(p + q)(1)(p + q)(2) , [tex]\int\limits^2_ {-1} \,[/tex] [tex]x^{3}[/tex](p + q)(x) d(x) + c sin((p + q)(0))
= (3(p(4)+q(4)) + 5(p'(6)+q'(6))+b(p(1)+q(1))(p(2)+q(2)) , [tex]\int\limits^2_ {-1} \,[/tex] [tex]x^{3}[/tex](p(x)+q(x))d(x)+ c sin(p(0)+q(0)))
= (3([tex]\frac{\pi }{2}[/tex] +[tex]\frac{\pi }{2}[/tex])+5(0+0)+b([tex]\frac{\pi }{2}[/tex]+[tex]\frac{\pi }{2}[/tex])([tex]\frac{\pi }{2}[/tex]+[tex]\frac{\pi }{2}[/tex]), [tex]\int\limits^2_ {-1} \,[/tex] [tex]x^{3}[/tex]([tex]\frac{\pi }{2}[/tex]+[tex]\frac{\pi }{2}[/tex])d(x)+c sin([tex]\frac{\pi }{2}[/tex]+[tex]\frac{\pi }{2}[/tex]))
= (3([tex]\pi[/tex])+b[tex]\pi ^{2}[/tex],[tex]\frac{15\pi }{4}[/tex])
and
Tp + Tq = (3p(4) +5(p)'(6)+bp(1)(p)(2) , [tex]\int\limits^2_ {-1} \,[/tex] [tex]x^{3}[/tex](p)(x) d(x) + c sin((p)(0)) +(3(q)(4) +5(q)'(6)+b(q)(1)(q)(2) , [tex]\int\limits^2_ {-1} \,[/tex] [tex]x^{3}[/tex](q)(x) d(x) +c sin(q(0)))
= (3([tex]\frac{\pi }{2}[/tex])+5(0)+b([tex]\frac{\pi }{2}[/tex])([tex]\frac{\pi }{2}[/tex]), [tex]\int\limits^2_ {-1} \,[/tex] [tex]x^{3}[/tex]([tex]\frac{\pi }{2}[/tex])d(x)+c sin([tex]\frac{\pi }{2}[/tex])) + (3([tex]\frac{\pi }{2}[/tex])+5(0)+b([tex]\frac{\pi }{2}[/tex])([tex]\frac{\pi }{2}[/tex]), [tex]\int\limits^2_ {-1} \,[/tex] [tex]x^{3}[/tex]([tex]\frac{\pi }{2}[/tex])d(x)+c sin([tex]\frac{\pi }{2}[/tex]))
=(3([tex]\frac{\pi }{2}[/tex])+[tex]\frac{\pi b}{4}[/tex],[tex]\frac{\pi }{2}[/tex] [tex]\int\limits^2_ {-1} \,[/tex] [tex]x^{3}[/tex]d(x) + c) +(3([tex]\frac{\pi }{2}[/tex])+[tex]\frac{\pi b}{4}[/tex],[tex]\frac{\pi }{2}[/tex] [tex]\int\limits^2_ {-1} \,[/tex] [tex]x^{3}[/tex]d(x) + c)
=(3[tex]\pi[/tex]+ [tex]\frac{\pi b}{2}[/tex],[tex]\frac{15\pi }{4}[/tex]+2c)
Since T is linear, additivity of T holds and implies that we have
(3([tex]\pi[/tex])+b[tex]\pi ^{2}[/tex],[tex]\frac{15\pi }{4}[/tex]) = T(p+q)
=Tp+Tq
=(3[tex]\pi[/tex]+ [tex]\frac{\pi b}{2}[/tex],[tex]\frac{15\pi }{4}[/tex]+2c)
from which we can equate the coordinates to obtain the equations 3π + πb/2= 3π +πb/2 and 15π/4 =15π/4+2c, which imply b = 0 and c = 0, respectively.
Backward direction: If b = 0 and c = 0, then T is linear. Suppose b = 0 and c = 0. Then the map T : R ^3 → R^2 becomes
Tp = (3p(4) +5(p)'(6) , [tex]\int\limits^2_ {-1} \,[/tex] [tex]x^{3}[/tex](p)(x) d(x) )
we need to prove that T is linear
• Additivity: For all p, q ∈ P(R), we have
T(p+q) = (3(p + q)(4) +5(p + q)'(6) , [tex]\int\limits^2_ {-1} \,[/tex] [tex]x^{3}[/tex](p + q)(x) d(x))
=(3(p(4)+q(4)) + 5(p'(6)+q'(6)) , [tex]\int\limits^2_ {-1} \,[/tex] [tex]x^{3}[/tex](p(x)+q(x))d(x))
= (3p(4) +5(p)'(6)+3q(4)+5q'(6) , [tex]\int\limits^2_ {-1} \,[/tex] [tex]x^{3}[/tex](p)(x) d(x) + [tex]\int\limits^2_ {-1} \,[/tex] [tex]x^{3}[/tex]q(x) d(x))
= (3p(4) +5(p)'(6) , [tex]\int\limits^2_ {-1} \,[/tex] [tex]x^{3}[/tex](p)(x) d(x)) +(3(q)(4) +5(q)'(6), [tex]\int\limits^2_ {-1} \,[/tex] [tex]x^{3}[/tex](q)(x) d(x))
=Tp+Tq
• Homogeneity: For all λ ∈ F and for all (x, y, z) ∈ R^3, we have
T(λp) = (3(λp)(4) + 5(λp)'(6), [tex]\int\limits^2_ {-1} \,[/tex] [tex]x^{3}[/tex](λp)(x) d(x))
=(3λp(4) + 5λp'(6), [tex]\int\limits^2_ {-1} \,[/tex] [tex]x^{3}[/tex]λ(p)(x) d(x))
=(λ(3p(4) + 5p'(6)),λ [tex]\int\limits^2_ {-1} \,[/tex] [tex]x^{3}[/tex]p)(x) d(x))
=λ(3p(4) + 5p'(6), [tex]\int\limits^2_ {-1} \,[/tex] [tex]x^{3}[/tex]p)(x) d(x))
=λTp
Therefore, additivity and homogeneity of T are satisfied, T is linear.
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help please i don’t understand this!
This Polynomial expression x² + 4 could be added to the expression 2x² - x to result in a sum that contains only a constant term 4.
What is the polynomial expression?
Any expression which consists of variables, constants, and exponents, and is combined using mathematical operators like addition, subtraction, multiplication, and division is a polynomial expression. Polynomial expressions can be classified as monomials, binomials, and trinomials according to the number of terms present in the expression.
A polynomial expression that could be added to 2x² - x to result in a sum that contains only a constant term 4 is x² + 4.
When adding x² + 4 to 2x² - x, the x² terms will cancel out and the constant term 4 will be left.
(x² + 4) + (2x² - x) = (x² + 2x²) + (4 - x) = 3x² -x + 4
The resultant polynomial expression is 3x² -x + 4 in which the only constant term is 4.
Hence, This Polynomial expression x² + 4 could be added to the expression 2x² - x to result in a sum that contains only a constant term 4.
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