By the quotient rule the derivative of all functions can be found in an easy way,
(1) [tex]f'(x)=\frac{x^6+128x^3-3x^2}{(x^3+32)^2}[/tex]
(2) [tex]f'(x)=\frac{5}{(2+x)^2}[/tex]
(3) [tex]f'(t)=\frac{4t}{(t^2+1)^2}[/tex]
The Quotient Rule in calculus is a technique for figuring out a function's derivative (differentiation) as the ratio of two differentiable functions. It is a formal rule applied to issues involving differentiation where one function is split in half by another function. The definition of the limit of the derivative is followed by the quotient rule. Keep in mind that the bottom function is where the quotient rule starts and where it finishes.
The quotient rule is similar to the product rule. A Quotient Rule is stated as the ratio of the quantity of the denominator times the derivative of the numerator function minus the numerator times the derivative of the denominator function to the square of the denominator function. In short, the quotient rule is a way of differentiating the division of functions or the quotients
We have the functions
(1) [tex]f(x)=\frac{x^4+1}{x^3+32}[/tex]
derivative with respect to x by the quotient rule.
[tex]f'(x)=\frac{(x^3+32)(4x^3)-(x^4+1)(3x^2)}{(x^3+32)^2}\\\\f'(x)=\frac{4x^6+128x^3-3x^6-3x^2}{(x^3+32)^2}\\\\f'(x)=\frac{x^6+128x^3-3x^2}{(x^3+32)^2}[/tex]
2) derivative with respect to x of the below function
[tex]f(x)=\frac{3x+1}{2+x}\\\\f'(x)=\frac{(x+2)(3)-((3x+1))}{(2+x)^2}\\\\f'(x)=\frac{5}{(2+x)^2}[/tex]
3)
[tex]f(t)=\frac{t^2-1}{t^2+1}[/tex]
derivative with respect to t,
[tex]f'(t)=\frac{(t^2+1)(2t)-(t^2-1)(2t)}{(t^2+1)}\\\\f'(t)=(2t^3+2t-2t^3+2t)/(t^2+1)^2\\\\f'(t)=4t/(t^2+1)^2[/tex]
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The following data show the ages of recent award-winning male actors at the time when they won their award. Make a frequency table for the data, using bins of 20-29, 30-39, and so on.
Click the icon to view the ages of male actors.
Complete the table below.
Age
20-29
30-39
40-49
50-59
60-69
70-79
No. of actors
Here is the frequency table for the data, using bins of 20-29, 30-39, and so on:
Age
20-29: 14 actors30-39: 14 actors40-49: 14 actors50-59: 7 actors60-69: 4 actors70-79: 1 actorWhat is a Frequency Table?A table that is created to display the distribution of a characteristic's frequency of occurrence in accordance with a specific set of class intervals.
With the ages and data sets given, we are able to compute the frequency of occurrence in the data set using the specific class interval given.
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27. A man is climbing down from 225 feet high descending 25 feet per minute. Write an
equation/function that models this situation.
PLEASE HELP
“Simplify the following using the properties of exponents”
A). (3x^-3y^5)^4
B). 5x^0
Answer:
[tex]A) 81x^{-12}y^{20}[/tex]
B) 5
Step-by-step explanation:
[tex]A) (3x^{-3}y^5)^4=(3)^4 \,\cdot\, (x^{-3})^4\,\cdot\,(y^5)^4=81x^{-12}y^{20}\\\\B) 5x^0=5(1)=5[/tex]
convert 10111.011 base 2 to binary
Answer:
Step-by-step explanation:
The number 10111.011 in base 2 is already in binary, so no conversion is necessary.
6-3) the turkalike rug company buys medium grade carpet in 100-foot rolls. the average number of defects per roll is 2.0. assuming that these data follow a poisson distribution, use the poisson spreadsheet template to answer the following questions. a. what is the probability of finding exactly 6 defects in a carpet roll chosen at random? b. what is the probability of finding 3 or fewer defects in carpet roll? using the poisson distribution.
The probability of finding exactly 6 defects in a carpet roll is 0.0435 and the probability of finding 2 or fewer defects in a carpet roll is 0.405.
a. To find the probability of finding exactly 6 defects in a carpet roll chosen at random, we can use the Poisson formula:
P(X = 6) = (e^-λ * λ ^6) / 6!
putting λ = 2.0, we get:
P(X = 6) = (e^-2 * 2^6) / 6! = 0.0435
So the probability of 6 defects in a carpet roll chosen at random is 0.0435.
b. To find the probability of finding 2 or fewer defects in a carpet roll, we can use the cumulative distribution function of the Poisson distribution:
P(X <= 2) = 1 - P(X > 2)
Using the Poisson formula, we can calculate P(X > 2) as follows:
P(X > 2) = 1 - (P(X = 0) + P(X = 1) + P(X = 2))
Plugging in λ = 2.0, we get:
P(X > 2) = 1 - (0.135 + 0.27 + 0.54) = 0.595
So P(X <= 2) = 1 - 0.595 = 0.405
So the probability of finding 2 or fewer defects in a carpet roll is 0.405.
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___The given question is incorrect, the correct question is given below:
The Turkalike Rug Company buys medium grade carpet in 100-foot rolls. The average number of defects per roll is 2.0. Assuming that these data follow a Poisson distribution, use the Poisson spreadsheet template to answer the following questions.
a) What is the probability of finding exactly 6 defects in a carpet roll chosen at random?
b) What is the probability of finding 2 or fewer defects in a carpet roll?
An investment firm offers its customers municipal bonds that mature after varying number of years. Given that the cumulative distribution function of t, the number of years to maturity for a randomly selected bond is:f(x) = { 0 t < 1{ 1/4 1 ≤ t < 3{ 1/2 3 ≤ t < 5{ 3/4 5 ≤ t < 7{ 1 t ≥ 7Find:a) P(T = 5)b) P(T > 3)c) P(1.4 < T < 6)d) P(T ≤ 5 | T ≥ 2)
The cumulative distribution function of t, the number of years to maturity for a randomly selected bond is a. 1/4, b. 3/4, c. 1/2, d. 1/3.
The probability of a randomly selected bond maturing exactly after 5 years is 1/4. The probability of a randomly selected bond maturing after more than 3 years is 3/4. The probability of a randomly selected bond maturing between 1.4 and 6 years is 1/2.
a) P(T=5) = F(5) - F(4) = 3/4 - 1/2 = 1/4
b) P(T>3) = 1 - P(T≤3) = 1 - F(3) = 1 - 1/4 = 3/4
c) P(1.4 < T < 6) = F(6) - F(1.4) = 3/4 - 1/4 = 1/2
d) P(T≤5 | T≥2) = P(2 ≤ T ≤ 5) / P(T≥2) = (F(5) - F(2)) / (1 - F(2)) = (3/4 - 1/4) / (1 - 1/4) = 1/3
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The question is -
An investment firm offers its customers municipal bonds that mature after varying numbers of years. Given that the cumulative distribution function of t, the number of years to maturity for a randomly selected bond is:
f ( x ) = ⎧ 0 t < 1
⎪ 1 / 4 1 ≤ t < 3
⎪ 1 / 2 3 ≤ t < 5
⎨ 3 / 4 5 ≤ t < 7
⎩ 1 t ≥ 7
Find: a) P(T = 5) b) P(T > 3) c) P(1.4 < T < 6) d) P(T ≤ 5 | T ≥ 2)
Q. 2 Transmission Trials In a wireless automatic meter reading system, a base station sends out a wake up signal to nearby electric meters. Upon receiving the wake up signal, a meter transmits a message indicating the electric usage. This message is repeated 4 times. A single transmission of the message is error-free with probability
p=0.9
, independent of the other transmissions. 1. Let
N
be a random variable that denotes the number of error-free transmissions for a single meter upon receiving the wake-up call. Find the PMF of
N
2. In practice, the system does not know which messages are error-free. It uses an error correction scheme which is able to decode the correct message if at least 3 out of 4 transmissions are error-free. What is the probability that the system is able to decode the correct message?
The probability that the system can decode the correct message is approximately 0.9477.
The random variable N denotes the number of error-free transmissions for a single meter upon receiving the wake-up call.
Since each transmission is error-free with probability p=0.9 and the transmissions are independent, the number of error-free transmissions follows a binomial distribution with parameters n=4 (number of trials) and p=0.9 (probability of success in each trial).
The probability mass function (PMF) of N is given by:
P(N=k) = (4 choose k) × [tex]0.9^k[/tex] × [tex]0.1^{(4-k)}[/tex], for k = 0, 1, 2, 3, 4
where (4 choose k) is the binomial coefficient, which represents the number of ways to choose k successes out of 4 trials.
The error correction scheme can decode the correct message if at least 3 out of 4 transmissions are error-free. Let X be the number of error-free transmissions for a single meter.
Then the probability that the system can decode the correct message is:
P(X >= 3) = P(X = 3) + P(X = 4)
To find these probabilities, we use the PMF of N from part 1:
P(X = 3) = P(N = 3) + P(N = 4) = (4 choose 3) × [tex]0.9^3[/tex] × 0.1 + (4 choose 4) × [tex]0.9^4[/tex] × [tex]0.1^0[/tex] = 0.2916
P(X = 4) = P(N = 4) = (4 choose 4) × [tex]0.9^4[/tex] × [tex]0.1^0[/tex] = 0.6561
Therefore, the probability that the system can decode the correct message is:
P(X >= 3) = P(X = 3) + P(X = 4) = 0.2916 + 0.6561 = 0.9477.
So the probability that the system can decode the correct message is approximately 0.9477.
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Trig ratios, pls hurry
The person is 36m away from the boat.
Applying simultaneous equations to solve angle of depressionThe given figure is triangular in nature. In order to determine the distance of the person from the boat, we will use the pythagoras theorem.
Given the following parameters
GH = 20m
GK = 30m
The measure of the length HK (The distance of the person from the boat,) is calculated as:
HK^2 = GH^2 + GK^2
HK^2 = 20^2+ 30^2
HK^2 = 400+900
HK^2 = 1300
HK = 36m
Hence the distance of the person from the boat is approximately 36m
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Please help me urgently
Answer:
Below
Step-by-step explanation:
25 oo pass / hr
# passengers for H hours would be 2500H
f (H) = 2500 H
Answer:
a) 2500H
b) 20,000
Step-by-step explanation:
Algebraic equation:To find the number of passengers carried, multiply the number of passengers carried in an hour by the total hours.
a) Passengers carried in an hour = 2500
Passengers carried in 'H' hours = 2500 * H = 2500H
b) H = 8 hours
Passengers carried in 8 hours = 2500 * 8
= 20,000
Li Marie's water tables are evenly spaced along the 5,000-meter (or 5-kilometer) course, including one at the beginning and one at the end. What is the distance between each pair of tables that are next to each other? Hint: There are not 9 sections between tables.
The distance between each of the 9 tables is given as follows:
625 meters.
How to obtain the distance?The distance is obtained applying the proportions in the context of the problem.
A proportion is applied as the distance is calculated by the division of the total distance by the number of spaces, which is one less than the number of tables.
The parameters for this problem are given as follows:
Total distance of 5000 meters.Eight spaces.Hence the distance between each of the 9 tables is obtained as follows:
5000/8 = 625 meters.
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Determine the lateral area and the surface area of each triangular prism by determining the area of the shape’s net.
Btw: They are all 1
The lateral area of the triangular prism is 3 cm².
The surface area of the triangular prism is 4 cm².
How to find the lateral and surface area of a triangular prism?The diagram above is a triangular prism. The lateral area can be found as follows:
lateral area of the triangular prism = perimeter of the base × height
Therefore,
lateral area of the triangular prism = (1 + 1 + 1) × 1
lateral area of the triangular prism = 3 × 1
lateral area of the triangular prism = 3 cm²
Therefore,
surface area of the triangular prism = perimeter of the base × height + 2(base area)
Therefore,
surface area of the triangular prism = 3 + (1 × 1)
surface area of the triangular prism = 4 cm²
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The variables x and y are connected by the equation y=(x-3)(x - 5)(x+2).Some corresponding values of x and y are shown in the table below. X -2 0 1 y 0 30 24 (i) Calculate the value of k -1 24 2 12 3 0 4 -6 5 0 (0) 6 k [1] avis
The value of k is given as follows:
k = 24.
How to find the numeric value of a function or of an expression?To find the numeric value of a function or of an expression, we replace each instance of the variable in the function or in the expression by the value at which we want to find the numeric value.
From the table, we have that when x = 6, y = k, hence the value of k is found replacing each of the three instances of x by 6, as follows:
k = (6 - 3)(6 - 5)(6 + 2)
k = 3 x 1 x 8
k = 24.
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Math help needed, within 30 minutes please :)
34.) If a fruit punch is made using 3 parts strawberries, 2 parts blueberry, 2 parts lime juice, 3 parts simple syrup, and 6 parts I’ve. How many ounces of each ingredient do you need to make five gallons? (Note: 1 gal=128 oz.)
Show with steps please! It’s a practice problem and I’m trying to learn. :)
120 ounce of strawberry, 80 ounce of blueberry, 80 ounce of lime juice, 120 ounce of simple syrup and 240 ounce of sugar is needed to make five gallons (640 ounces) of fruit punch
What is an equation?An equation is an expression that shows how numbers and variables are linked together using mathematical operations such as addition, subtraction, multiplication and division.
1 gal = 128 oz. Hence:
5 gal = 5 gal * 128 oz. per gal = 640 ounces
If a fruit punch is made using 3 parts strawberries, 2 parts blueberry, 2 parts lime juice, 3 parts simple syrup, and 6 parts sugar. Hence. of 5 gallons:
Amount of strawberry = [3/(3 + 2 + 2 + 3 + 6)] * 640 ounces = (3/16) * 640 = 120 ounces
Amount of blueberry = [2/(3 + 2 + 2 + 3 + 6)] * 640 ounces = (2/16) * 640 = 80 ounces
Amount of lime juice = [2/(3 + 2 + 2 + 3 + 6)] * 640 ounces = (2/16) * 640 = 80 ounces
Amount of simple syrup = [3/(3 + 2 + 2 + 3 + 6)] * 640 ounces = (3/16) * 640 = 120 ounces
Amount of sugar = [6/(3 + 2 + 2 + 3 + 6)] * 640 ounces = (6/16) * 640 = 240 ounces
120 ounce of strawberry, 80 ounce of blueberry, 80 ounce of lime juice, 120 ounce of simple syrup and 240 ounce of sugar is needed
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1
A reforestation drive is conducted in a town. The town had 2,022 trees at the start of the drive. Four months after the start of the
reforestation drive, the town had 4,897 trees.
If n represents the number of months and T represents the number of trees, which of the following equations can be used to model
this situation?
OA. T 718.75 - 2,022
OB. T = 2,022n + 718.75
OC. T 718.75 + 2,022
OD. T = 2875 + 2,022n
Answer: OD. T = 2875 + 2,022n
Step-by-step explanation: What makes the most sense is
OD. T = 2875 + 2,022n
because they added 2875 to the trees already there and 4,897 was the new total after four months, I also want to point out it has nothing to do with decimals or a 700 number.
ANSWER ASAP. 50PTS. BRAINLIEST ANSWER.
What is the area of Triangle PQR on the grid?
A triangle PQR is shown on a grid. The vertex P is on ordered pair 1 and 4, vertex Q is on ordered pair 3 and 1, and the vertex R is on ordered pair 5 and 4.
(1 point)
5 square units
6 square units
10 square units
12 square units
The Area of given triangle is 8 square units.
Area of Triangle:Area of triangle is the total area occupied by the triangle or the area enclosed within the three sides of the triangle.
The formula for calculating area of triangle is,
A = 1/2 bh
where b and h represent the base and height of the triangle, respectively.
This formula can be used to calculate triangles of any shape, including scalene, isosceles, and equilateral triangles. Remember that the base and height of a triangle are perpendicular to one another.
Now in the given question,
P(1,4), Q(3,1), R(5,4)
We have PR parallel to the x axis. We'll call that the base,
b = 5 - 1 = 4
The altitude is then the y difference h = 5 - 1 = 4
Then area of given triangle is,
[tex]A=\frac{1}{2}bh\\\\\\A=\frac{1}{2} (4)(4)\\\\A=\frac{1}{2} (16)\\\\A=8[/tex]
Hence, the area of given triangle is 8 square units.
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Solve and explain how you would use ratios and proportions to answer
the following two questions:
If Layla paid $48 for 4 movie tickets, how much would it cost her to
purchase 7 tickets? How much does each ticket cost? !USE RATIOS AND PROPORTIONS IN YOUR ANSWER!
Answer:
1/ $12 for each ticket
2/ $84 for 7 tickets
Step-by-step explanation:
We know
Layla paid $48 for 4 movie tickets
How much does each ticket cost?
48 / 4 = $12 for each ticket
So, each ticket cost $12
How much would it cost her to purchase 7 tickets?
12 x 7 = $84
So, it cost her $84 to purchase 7 tickets.
2. By using a truthtable determine whether the following argument is valid or not If 10,836 is divisible by 12 then 10836 is divisible by 3. If 10,836 is divisible by 3 then the sum of the digits of 10836 is divisible by 3. Therefore, if 10,836 is divisible by 12 then the sum of the digits of 10836 is divisible by 3. (10 marks)
Based on the truth table constructed, the premises and conclusions are true, hence, the argument is valid.
What is the validity of the argument?The validity of the argument is determined as follows:
First statement, S1:
10836/12 = 903
10836/3 = 3612
Second statement, S2:
10836/3 = 3612
1 + 0 + 8 + 3 + 6 = 18
18/3 = 6
Statement 3, S3:
10836/12 = 903
1 + 0 + 8 + 3 + 6 = 18
18/3 = 6
The truth table is given below
Premise 1 Premise 2 Conclusion
S1. T T T
S2. T T T
S3. T T T
Since all the premises and conclusions are true, the argument is valid.
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Listed below are brain volumes (cm3) of unrelated subjects used in a study. Use the sample data to construct a 99% confidence interval estimate of the mean of the brain volume of the population. 963 1027 1272 1079 1070 1173 1067 1347 1100 1204
The 99% confidence interval estimate of the mean of the brain volume of the population is given as follows:
(1009.5, 1250.9).
What is a t-distribution confidence interval?The t-distribution is used when the standard deviation for the population is not known, and the bounds of the confidence interval are given according to the following rule:
[tex]\overline{x} \pm t\frac{s}{\sqrt{n}}[/tex]
In which the variables of the equation are presented as follows:
[tex]\overline{x}[/tex] is the sample mean.t is the critical value.n is the sample size.s is the standard deviation for the sample.The critical value, using a t-distribution calculator, for a two-tailed 99% confidence interval, with 10 - 1 = 9 df, is t = 3.2498.
Using a calculator, the parameters are given as follows:
[tex]\overline{x} = 1130.2, s = 117.45, n = 10[/tex]
The lower bound of the interval is of:
1130.2 - 3.2498 x 117.45/sqrt(10) = 1009.5.
The upper bound of the interval is of:
1130.2 + 3.2498 x 117.45/sqrt(10) = 1250.9.
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Admission to a circus is $16 for adults and $8 for children Select the numerical expression that shows the total cost of 3 adults and 8 children.
24
⋅
11
24⋅11
16
⋅
3
+
8
⋅
8
16⋅3+8⋅8
16
+
8
⋅
3
⋅
8
16+8⋅3⋅8
Answer:b or 2 option
Step-by-step explanation= $16x3=48 and $8x8=64 adding those together equals $112 and that is what answer b says i would really apreaciate branliest.
in a recent research poll, 27 of 90 randomly selected men in the u.s. were able to identify india on a map. when 100 randomly selected women in the u.s. were asked, 38 were able to identify india on a map. construct and interpret a 95% confidence interval for the true difference in the proportion of u.s. men and the proportion of u.s. women who can identify india on a map.STATE:99% CI for _____ where______ = The _______of all U.S. men who can identify India on the map and ______= the______ of all US women who can identify India on the map,
State: 99% CI for the true difference in proportions where p1 = the proportion of all U.S. men who can identify India on the map and p2 = the proportion of all US women who can identify India on the map. The main answer is (-0.216, 0.056).
The 95% confidence interval for the true difference in the proportion of U.S. men and women who can identify India on a map is (-0.206, 0.066). This means that we are 95% confident that the true difference in proportions lies between -0.206 and 0.066.
To construct this confidence interval, we use the formula:
CI = (p1 - p2) ± z*SE
where p1 and p2 are the sample proportions for men and women respectively, z is the critical value for a 95% confidence level (1.96), and SE is the standard error of the difference in proportions.
The sample proportion for men is 27/90 = 0.3, and for women it is 38/100 = 0.38. Therefore, p1 - p2 = -0.08. The standard error is:
SE = sqrt(p1*(1-p1)/n1 + p2*(1-p2)/n2) = sqrt(0.30.7/90 + 0.380.62/100) = 0.084
Substituting these values into the formula, we get:
CI = (-0.08) ± 1.96*0.084 = (-0.206, 0.066)
Thus, we can say with 95% confidence that the true difference in proportions of U.S. men and women who can identify India on a map lies between -0.206 and 0.066. This means that there may not be a significant difference between the two proportions.
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in terms of the function s, the limit we would have to evaluate in order to calculate the slope of the tangent line to s at t=1 is
It is not possible to give an exact answer to this question without knowing the specific function s. However, in general, the slope of the tangent line to a function s at a specific point t can be found by taking the derivative of the function s at that point.
If we let f(t) be the function s, then the derivative of f(t) at t=1 would give us the slope of the tangent line to s at that point. This can be written mathematically as:
s'(1) = lim(h->0) [s(1+h) - s(1)]/h
Here, s'(1) represents the derivative of s at t=1, and h represents a small change in t. By taking the limit as h approaches 0, we can find the instantaneous rate of change of s at t=1, which is the slope of the tangent line to s at that point.
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A line passes through the point (-4, 6) and has a slope of 5/4. Write and equation in slope-intercept form for this line
In slope-intercept form, the equation of the line is y = (5/4)x + 6 1/2.
What is equation?An equation in mathematics is a statement that two expressions are equal. An equation can be written in the form of a mathematical expression on the left side equal to a value or another mathematical expression on the right side. There are different types of equations, including linear equations, quadratic equations, and exponential equations, among others. Equations can be solved for a specific variable, and the solution is the value of the variable that makes the equation true. The process of solving an equation involves isolating the variable on one side of the equation and finding its value.
Here,
The slope-intercept form of a line is given by y = mx + b, where m is the slope of the line and b is the y-intercept, which is the point at which the line intersects the y-axis. To find the equation of a line in slope-intercept form given its slope and a point on the line, we can use the following steps:
Use the point-slope form of a line: y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
Plug in the slope and the point: y - 6 = (5/4)(x + 4).
Solve for y: y = (5/4)x + (26/4).
Simplify the equation: y = (5/4)x + 6 1/2.
So, the equation of the line in slope-intercept form is y = (5/4)x + 6 1/2.
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Paying into a retirement savings plan will increase taxable income, true or false
Answer:
true
Step-by-step explanation:
How do I solve y = 2x + 7
y=-3x+17
Answer:
Make the expressions equal to each other.
2x + 7 = -3x + 17
Use subtraction, division, and addition to isolate the variable.
2x + 3x + 7 = -3x + 3x + 17
5x + 7 = 17
5x + 7 - 7 = 17 - 7
5x = 10
5x/5 = 10/5
x = 2
y = 11
Step-by-step explanation:
Which relation is a function?
The relation that is a function is D (see image below).
How to Determine if a Relation is a Function Using the Vertical Line Test?If any vertical line passes through more than one point on a graph, then the relation is not a function. Conversely, if every vertical line intersects the graph at most one point, then the relation is a function.
Applying the vertical line test, if we place a vertical line across each graph given, only the graph in option D (as shown in the image below) will have the the vertical line intersects it at most one point.
Therefore, the correct option is D. (see image below).
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A photograph measuring 4" wide × 6" long must be reduced in size to fit a space four inches long in an advertising brochure. How wide must the space be so that the picture remains in proportion?
Answer:
The proportion of the width to the length of the original photograph is 4:6, which can be simplified to 2:3. To maintain the same proportion, the width of the reduced photograph must also be 2/3 of the length.
Since the length of the space in the brochure is 4 inches, the width of the space must be:
width = (2/3) x length
width = (2/3) x 4
width = 8/3
width ≈ 2.67 inches
Therefore, the width of the space in the brochure should be approximately 2.67 inches to maintain the proportion of the photograph.
Referring to the Fig. in Question #23, find the cosine of angle S. Reduce the answer to the lowest terms.
The Cosine of angle S in the given triangle is
[tex]\frac{8}{10} =\frac{4}{5}[/tex] .
Trigonometric Identities:Trigonometric Identities are equality claims that apply the principles of trigonometry and are valid for all possible values of the variables in the equation.
Many particular trigonometric identities can be used to express the relationship between a triangle's side length and angle.Only the right-angle triangle is consistent with the trigonometric identities.
All trigonometric identities are built upon the foundation of the six trigonometric ratios. Some of their names are sine, cosine, tangent, cosecant, secant, and cotangent. Each of these trigonometric ratios is defined using the adjacent side, opposite side, and hypotenuse side of the right triangle. All fundamental trigonometric identities are derived from the six trigonometric ratios.
The Cosine of an angle ∅ in a right triangle is given as ,
Cosine (∅) = [tex]\frac{base}{hypotenuse}[/tex]
Now in the given triangle .
[tex]Cosine (S)=\frac{8}{10} =\frac{4}{5}[/tex]
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Which cross sections is possible in cylinders and cones
The cross-section of the cylinders and cones is always round.
What is a cross-section area?The cross-sectional area is the region of a two-dimensional shape produced when a three-dimensional object, such as a cylinder, is cut perpendicular to a preset axis at a point.
A cylinder is one of the most basic curvilinear geometric shapes and has traditionally been solid in three dimensions. In elementary geometry, it is regarded as a prism with a circle as its basis.
A three-dimensional geometric shape known as a cone has a flat base and a smooth, tapering vertex. A cone is made up of several line segments, half-lines, or lines that connect to a single point.
As a result, the cross sections of the cones and cylinders are always round.
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Consider the curve C parametrized by x= t and y (`6-t^2)^1/2 for -4≤ t ≤ 4 write an equation relating x and y but without t and without square roots. write it so variable terms are on one side and any constant on the other side.
An equation relating x and y but without t and square roots is represented by x² + y² = 16, here variable terms are present on one side and any constant on the other side.
A function is represented in the cartesian form, such as, f(x,y)=0 or expressed in the form of parametric equations, written as
r(t) = ⟨x(t),y(t)⟩, α ≤ t ≤ β
the position vector, x(t) and y(t) helps to determine the location of a particular point on the curve with respect to a third independent parameter or varible t. We have, a curve C with parametrized equation, x = t and y = √(6-t²) for -4≤ t ≤ 4. We have to write eqution related to x and y without t and square roots. So,
substitute t = x in y
=> y = √(16 − x²)
squaring on both sides
=> y² = (16 - x²)
=> x² + y² = 16
Hence, the required equation is x² + y² = 16.
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Calculating angle of elevation of sun at different times of the day using your height and the length of shadow cast
At noon on a clear day, with a height of 6 feet and a length of the shadow of 6 feet, the angle of elevation of the sun is approximately 45 degrees.
What is the angle of elevation?An angle of elevation is defined as the angle formed between the horizontal plane and the line of sight from the observer's eye to anything above.
Let's assume a height of 6 feet and a length of the shadow of 6 feet.
We'll also assume that we are measuring the angle of elevation at noon on a clear day.
Using the tangent formula, we have:
tan θ = h / s
tan θ = 6/6
θ = tan⁻¹(6 / 6)
θ = tan⁻¹(1)
θ = 45°
Therefore, the angle of elevation of the sun is approximately 45 degrees.
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