We need to use the remainder theorem to find the remainder when the polynomial p(x) = x^4 - 9x^3 - 5x^2 - 3x + 4 is divided by the polynomial x - 3. The remainder when p(x) is divided by x - 3 is -212.
The remainder theorem states that if a polynomial f(x) is divided by x - a, then the remainder is equal to f(a).
In this case, we want to find the remainder when p(x) is divided by x - 3. To do this, we substitute x = 3 into the polynomial p(x) and calculate the result.
p(3) = (3)^4 - 9(3)^3 - 5(3)^2 - 3(3) + 4
= 81 - 243 - 45 - 9 + 4
= -212
Therefore, the remainder when p(x) is divided by x - 3 is -212.
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What is the probability of 3 people sharing the same birthdays? How many different pairs of people are there when there are three humans? (Think nPr or nCr)
The probability of three people sharing the same birthdays is approximately [tex]0.0000075[/tex] or [tex]0.00075[/tex]%, and there are three different pairs of people when there are three humans.
The probability of three people sharing the same birthday depends on the assumptions made about the distribution of birthdays. Assuming that birthdays are uniformly distributed throughout the year and that leap years are not considered, there are [tex]365[/tex] possible birthdays for each person. The first person can have any birthday, and the probability that the second person shares the same birthday is [tex]$\frac{1}{365}$[/tex]. Similarly, the probability that the third person shares the same birthday as the first two is also [tex]$\frac{1}{365}$[/tex]. Multiplying these probabilities together, we get [tex]$\left(\frac{1}{365}\right) \times \left(\frac{1}{365}\right) = \frac{1}{133,225}$[/tex], approximately [tex]0.0000075[/tex] or [tex]0.00075\%[/tex].When there are three humans, the number of different pairs of people can be calculated using the combination formula, also known as [tex]$\binom{n}{r}$[/tex]. In this case, [tex]$n$[/tex] represents the total number of people ([tex]3[/tex]), and [tex]$r$[/tex] represents the number of people chosen at a time ([tex]2[/tex] for pairs). Applying the formula, we have [tex]$\binom{3}{2} = 3$[/tex]. Therefore, there are three different pairs of people when there are three humans: ([tex]1,2[/tex]), ([tex]1,3[/tex]), and ([tex]2,3[/tex]).In conclusion, the probability of three people sharing the same birthdays is extremely low (approximately [tex]0.0000075 \ or \ 0.00075\%[/tex]), and when there are three humans, there exist three different pairs of people.
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what is the measure of angle A?
Answer:
122
Step-by-step explanation:
180 - (43 + 15)
Answer:
The answer is 2.86
Step-by-step explanation:
you just have to divide 43 by 15
I’m not sure how to solve this problem
Answer:
a
Step-by-step explanation:
The average car decreases in value by about 15% per year. If a car's original value is $28,000, which function best represents its value, y, after t years?
A. y=28,000(1+15)^t
B. y=28,000(1+0.15)^t
C. y=28,000(1-15)^t
D. y=28,000(1-0.15)^t
Answer:
D
Step-by-step explanation:
D
please help find What is AB?
Answer:
oblique
Step-by-step explanation:
please help with this ?!?
Simplify 7a - 3(b - a)
Answer:
10a-3b is your answer
Step-by-step explanation:
7a - 3(b - a)
7a-3b+3a
10a-3b
Help me please!! If you do you will get 25 points :)
Answer:
24 units by 15 units
Step-by-step explanation: To find how many units the length and width are, divide each by 5:
120/5 = 24
75/5= 15
For every 5 feet, there is 1 unit .
A car worth $14,000 depreciates at a rate of 4% per month. How long until it is worth $10,000?
Codification and Decodification let F = Z2. Consider the code
C = {000000, 001111, 110011, 111100, 101010}.
(a) Show that C is not a linear code.
b) Add words to C to form a new code C' that is linear.
c) Find a base of C'
Main Answer: The base of C' is {0110, 1001, 1100, 0011}.
Supporting Explanation: In a communication system, codification and decodification are used to encode and decode messages. C is the code for the message, where C={0000, 1100, 1010, 0110, 0101, 0011, 1001, 1111}. The code is a binary code since F=Z2. C' is the dual code of C. The codewords in C' are orthogonal to those in C. A basis for C' can be determined by finding a generator matrix for C'. Thus, the generator matrix for C is the parity check matrix for C'. A generator matrix for C is given as, G = [I | P] where P is the parity check matrix. The parity check matrix for C can be determined as, P = [-AT | Im-k]. Therefore, P = [0101; 1010; 1111].The rows of C' correspond to the columns of P. Thus, a basis for C' is {0110, 1001, 1100, 0011}.
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In ΔGHI, h = 650 cm, i = 130 cm and ∠G=72°. Find the area of ΔGHI, to the nearest square centimeter.
Answer:
84500 is the correct answer
Answer:
40182 delta math
Step-by-step explanation:
Calculate the following limits using the limit laws and limx→2f(x)=−3, limx→2g(x)=4, limx→2h(x)=7 (a) limx→2(f(x)−2g(x))= (b) limx→2(h(x)2)= (c) limx→2h(x)⋅g(x)f(x)=
The value of limits after using limit laws is [tex]$\lim_{x \to 2} \frac{h(x) \cdot g(x)}{f(x)} = -\frac{28}{3}$.[/tex]
What are Limit Laws?
Limit laws, also known as limit properties or limit theorems, are a set of rules and principles that allow us to simplify and evaluate limits of functions. These laws provide a systematic approach to finding the limit of a more complex expression by breaking it down into simpler parts.
Given:
[tex]\lim_{x \to 2} f(x) &= -3 \\\lim_{x \to 2} g(x) &= 4 \\\lim_{x \to 2} h(x) &= 7\end{align*}\textbf{(a) Calculate} $\lim_{x \to 2} (f(x) - 2g(x))$:[/tex]
Using the limit laws, we can split the expression and apply the limit laws individually:
[tex]\lim_{x \to 2} (f(x) - 2g(x)) &= \lim_{x \to 2} f(x) - \lim_{x \to 2} (2g(x)) \\&= \lim_{x \to 2} f(x) - 2 \lim_{x \to 2} g(x) \\&= (-3) - 2(4) \\&= -3 - 8 \\&= -11[/tex]
Therefore,[tex]$\lim_{x \to 2} (f(x) - 2g(x)) = -11$.[/tex]
[tex]\textbf{(b) Calculate} $\lim_{x \to 2} (h(x))^2$:[/tex]
Again, using the limit laws, we can apply the limit to the expression:
[tex]\lim_{x \to 2} (h(x))^2 &= \left(\lim_{x \to 2} h(x)\right)^2 \\&= (7)^2 \\&= 49[/tex]
Therefore,
[tex]\lim_{x \to 2} (h(x))^2 = 49$.\textbf{\\\\(c) Calculate} $\lim_{x \to 2} \frac{h(x) \cdot g(x)}{f(x)}$:[/tex]
Applying the limit laws, we can evaluate the limit as follows:
[tex]\lim_{x \to 2} \frac{h(x) \cdot g(x)}{f(x)} &= \frac{\lim_{x \to 2} h(x) \cdot \lim_{x \to 2} g(x)}{\lim_{x \to 2} f(x)} \\\\&= \frac{7 \cdot 4}{-3}\\ \\&= \frac{28}{-3}[/tex]
Therefore,[tex]$\lim_{x \to 2} \frac{h(x) \cdot g(x)}{f(x)} = -\frac{28}{3}$.[/tex]
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PLEASE ANSWER ALL
What is the equation of the axis of symmetry of the function?
What are the coordinates of the vertex of the function?
What are the coordinates of the x¬-intercepts of the function?
What are the coordinates of the y-intercept of the function?
Step-by-step explanation:
The axis of symmetry: is the line that makes the parabola split in exactly half and lines up with the vertex. For that parabola x=1 is the line of symetry.
The vertex is where the minimum of the graph is, on this graph you can eyeball it to be (1,-9)
The x-intercept is where y is 0 so that's where the lines intersex with the x-axis. (-2,0) and (4,0)
The y-intercept of the function is where x is 0 and where the parabola intersects with the y-axis. On this graph it would be (0,-8)
Hope that helps :)
1) Suppose a random variable X can only take the six values (1,2,3,4,5, and 6 ). If each value has equal probability, what is its pmf? b) Suppose the probabilities of X(0,1,2, and 3) are 1/9,2/9,2/9, and 4/9. show its pmf?
Answer : a) a random variable X can only take the six values (1,2,3,4,5, and 6 ). If each value has equal probability, then p(1) = p(2) = p(3) = p(4) = p(5) = p(6) = 1/6
b) The pmf of the random variable X is:p(0) = 1/9p(1) = 2/9p(2) = 2/9p(3) = 4/9
Explanation :
A probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value.[1] Sometimes it is also known as the discrete density function.
The probability mass function is often the primary means of defining a discrete probability distribution, and such functions exist for either scalar or multivariate random variables whose domain is discrete.
a) If each value has equal probability, the pmf of the random variable X which can only take the six values (1,2,3,4,5, and 6) is : p(1) = p(2) = p(3) = p(4) = p(5) = p(6) = 1/6
b)If the probabilities of X(0,1,2, and 3) are 1/9,2/9,2/9, and 4/9. The pmf of the random variable X is:p(0) = 1/9p(1) = 2/9p(2) = 2/9p(3) = 4/9
The sum of these probabilities is:p(0) + p(1) + p(2) + p(3) = 1/9 + 2/9 + 2/9 + 4/9 = 9/9 = 1
So, the pmf is defined for all X.
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Given f(x)=9+x and g(x)=3x-2, evaluate: fg(x)
Answer:
(f*g)(x) = 3x^2 + 25x - 18
Step-by-step explanation:
(f*g)(x) represents the product of the two functions f and g:
(f*g)(x) = 27x - 18 + 3x^2 - 2x, or, after simplification,
(f*g)(x) = 25x - 18 + 3x^2, or
(f*g)(x) = 3x^2 + 25x - 18
The distribution of actual weights of wedges of cheddar cheese produced at a dairy is normal with a mean of 10.2 ounces and a standard deviation of 0.2 ounces. (Round all answers to 4 decimal places, if needed.)
(a) The probability that a randomly chosen wedge of cheddar cheese is greater than 10.14 is .
(b) If a sample of 16 is randomly chosen, then the distribution of the sample mean weight is approximately normal not normal left-skewed right-skewed with a mean of ? and a standard deviation of .
(c) The probability that the sample mean weight of this sample of 16 is less than 10.14 is .
(d) The probability that the sample mean weight of this sample of 16 is greater than 10.14 is .
(e) The probability that the sample mean weight of this sample of 16 is between 10.14 and 10.3 is .
(f) There is only a 7% chance that the average weight of a sample of these 16 cheese wedges will be below .
(a) The probability that a randomly chosen wedge of cheddar cheese is greater than 10.14 is found using the standard normal distribution as follows:
P(Z > z) = P(Z > (10.14 - µ)/σ)
= P(Z > (10.14 - 10.2)/0.2)
≈ 0.3085.
Therefore, the probability is approximately 0.3085.
(b) If a sample of 16 is randomly chosen, then the distribution of the sample mean weight is approximately normal with a mean of 10.2 ounces and a standard deviation of σ/√n,
Where n = 16.
The sample standard deviation is given by σ = 0.2, so the standard deviation of the sample mean weight is:
σ/√n = 0.2/√16
= 0.05.
Therefore, the distribution of the sample mean weight is approximately normal with a mean of 10.2 ounces and a standard deviation of 0.05 ounces.
(c) The probability that the sample mean weight of this sample of 16 is less than 10.14 is found using the standard normal distribution as follows:
P(Z < z) = P(Z < (10.14 - µ)/(σ/√n))
= P(Z < (10.14 - 10.2)/(0.2/√16))
≈ P(Z < -1.6)
≈ 0.0548.
Therefore, the probability is approximately 0.0548.
(d) The probability that the sample mean weight of this sample of 16 is greater than 10.14 is found using the standard normal distribution as follows:
P(Z > z) = P(Z > (10.14 - µ)/(σ/√n))
= P(Z > (10.14 - 10.2)/(0.2/√16))
≈ P(Z > -1.6)
≈ 0.9452.
Therefore, the probability is approximately 0.9452.
(e) The probability that the sample mean weight of this sample of 16 is between 10.14 and 10.3 is found
Using the standard normal distribution as follows:
P(a < Z < b) = P((a - µ)/(σ/√n) < Z < (b - µ)/(σ/√n))
= P((10.14 - 10.2)/(0.2/√16) < Z < (10.3 - 10.2)/(0.2/√16))
≈ P(-1.6 < Z < 2)
≈ 0.9452 - 0.0548
= 0.8904.
Therefore, the probability is approximately 0.8904.
(f) Let x be the average weight of a sample of these 16 cheese wedges that is below some value z.
Then, the probability that x is less than z is 0.07.
Using the standard normal distribution, we can find the z-score such that
P(Z < z) = 0.07 as follows:
z = inv Norm(0.07)
≈ -1.4758.
Therefore, the average weight of a sample of these 16 cheese wedges that is below the value z is:
x = µ + z(σ/√n)
= 10.2 + (-1.4758)(0.2/√16)
≈ 10.0625.
Therefore, there is only a 7% chance that the average weight of a sample of these 16 cheese wedges will be below 10.0625.
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Determine whether the following functions are injective, or surjective, or neither injective nor surjective.
a) f ∶ {a, b, c, d} → {1, 2, 3, 4, 5} given by f (a) = 2, f (b) = 1, f (c) = 3, f (d) = 5. Is f injective? Is f surjective?
b) f ∶ R → R by f (x) = x + 1. Is f injective? Is f surjective?
c) f ∶ Z × Z → Z by f (m, n) = m + n. Is f injective? Is f surjective?
d) f ∶ Z × Z → Z by f (m, n) = m2 + n 2 . Is f injective? Is f surjective?
a) The function f is not injective but is surjective.
b) The function f is injective and surjective.
c) The function f is not injective but is surjective.
d) The function f is not injective and not surjective.
a) The function f maps four elements from the domain {a, b, c, d} to five elements in the codomain {1, 2, 3, 4, 5}. Since there are more elements in the codomain than the domain, f cannot be injective. However, since every element in the codomain is mapped to by at least one element in the domain, f is surjective.
b) The function f(x) = x + 1 is a linear function that maps every real number to a unique real number. Hence, f is injective. Additionally, for every real number y, there exists x = y - 1 such that f(x) = y, meaning f is surjective.
c) The function f(m, n) = m + n maps pairs of integers from the domain Z × Z to integers in the codomain Z. Since there are infinitely many pairs that can result in the same sum, f cannot be injective. However, for every integer in the codomain, there exists at least one pair of integers in the domain whose sum is equal to it, making f surjective.
d) The function f(m, n) = m^2 + n^2 maps pairs of integers from the domain Z × Z to integers in the codomain Z. Since different pairs of integers can have the same sum of squares, f is not injective. Furthermore, there are integers in the codomain that cannot be obtained as a sum of squares, making f not surjective.
In summary, the injectivity and surjectivity of the given functions are as follows: a) not injective, surjective; b) injective, surjective; c) not injective, surjective; d) not injective, not surjective.
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Chris makes $25 an hour and is getting a 10% raise. What is her new
earning per hour?
Answer:
$27.50
Step-by-step explanation:
Hope this helps and have a great day!!!!
Step-by-step explanation:
Since his pay is increased by 10 percent, then you multiply 10% by 25 and then add that to 25.
25+10/100(25)=55/2
27.5
Hope that helps :)
A thermometer is taken from a room where the temperature is 21 degrees Celsius to the outdoors, where the temperature is 5 degrees Celsius. After one minute the thermometer reads 15 degrees Celsius.
(a) What will the reading on the thermometer be after 3 more minutes?
(b) When will the thermometer read 6 degrees Celsius?
degrees Celsius
a) the reading on the thermometer after 3 more minutes will be -3 degrees Celsius.
b) the thermometer will read 6 degrees Celsius after 1.5 minutes.
To solve the given problem, we can assume that the temperature change follows a linear pattern based on the given information.
(a) To find the reading on the thermometer after 3 more minutes, we need to determine the rate of temperature change per minute. From the initial reading of 21 degrees Celsius to the reading after one minute of 15 degrees Celsius, there was a temperature decrease of 6 degrees Celsius in one minute.
Therefore, the rate of temperature decrease is 6 degrees Celsius per minute. If this rate remains constant, after 3 more minutes, the thermometer will show a further temperature decrease of:
3 minutes * 6 degrees Celsius per minute = 18 degrees Celsius
Thus, the reading on the thermometer after 3 more minutes will be 15 degrees Celsius - 18 degrees Celsius = -3 degrees Celsius.
(b) To find when the thermometer will read 6 degrees Celsius, we need to determine the time it takes for the temperature to decrease from 15 degrees Celsius to 6 degrees Celsius.
The initial reading is 15 degrees Celsius, and the final desired reading is 6 degrees Celsius. Therefore, we need to calculate the time it takes for a temperature decrease of:
15 degrees Celsius - 6 degrees Celsius = 9 degrees Celsius
Since the rate of temperature decrease is 6 degrees Celsius per minute, we can set up the equation:
9 degrees Celsius = 6 degrees Celsius per minute * t minutes
Solving for t (the time it takes to reach 6 degrees Celsius):
t = 9 degrees Celsius / 6 degrees Celsius per minute = 1.5 minutes
Therefore, the thermometer will read 6 degrees Celsius after 1.5 minutes.
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2. Suppose 250 randomly selected people are surveyed to determine if they own a tablet. Of the 250 surveyed, 98 reported owning a tablet. Using a 95% confidence level, compute a confidence interval estimate for the true proportion of people who own tablets.
A. With 95% confidence, we say that the proportion of people who own tables is between 32% and 98%.
B. With 95% confidence, we say that the proportion of people who own tables is between 32% and 99%.
C. With 95% confidence, we say that the proportion of people who own tables is between 33% and 98%.
D. With 95% confidence, we say that the proportion of people who own tables is between 33% and 99%.
Solution:
Given that a random-sample of 250 people is surveyed to determine if they own a tablet, where 98 people own a tablet.
We have to find a confidence interval estimate for the true proportion of people who own tablets using a 95% confidence-level.
The formula to compute confidence interval estimate is given by;
[tex]CI = p \pm Z_{\frac{\alpha}{2}}\sqrt{\frac{p(1-p)}{n}}[/tex]
Where;[tex]p[/tex] = Sample proportion[tex]Z_{\frac{\alpha}{2}}[/tex] = Critical value of Z at [tex]\frac{\alpha}{2}[/tex][tex]n[/tex] = Sample size
From the given data,Sample proportion, [tex]p = \frac{98}{250} = 0.392[/tex]
Level of Confidence, [tex]C= 95%[/tex]
As level of significance [tex]\alpha = (1-C) = 0.05[/tex]So, [tex]\frac{\alpha}{2} = \frac{0.05}{2} = 0.025[/tex]
Sample size, [tex]n = 250[/tex]
Now, we need to find the critical value of [tex]Z_{0.025}[/tex] such that the area to its right in the z-distribution is 0.025.Z-table shows the values of Z for given probabilities.
The closest value to 0.025 is 1.96. So, we can take [tex]Z_{0.025} = 1.96[/tex].
Therefore, the confidence interval estimate for the true proportion of people who own tablets using a 95% confidence level is given as;[tex]CI = 0.392 \pm 1.96\sqrt{\frac{0.392(1-0.392)}{250}}[/tex][tex]\Rightarrow CI = 0.392 \pm 0.067[/tex]
So, the lower limit of the interval is obtained as;
[tex]0.392 - 0.067 = 0.325[/tex]
And the upper limit of the interval is obtained as;
[tex]0.392 + 0.067 = 0.459[/tex]
Therefore, with 95% confidence, we say that the proportion of people who own tablets is between 32.5% and 45.9%.
The correct option is (A).
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what are all possible values for x in the equation x^3=375?
Answer:
Select all possible values for x in the equation.
x cubed=375.
5*the cubed root of 3
the cubed root of 375
75*the cubed root of 5
125*the cubed root of 3
I am trying to do a practice test to prepare for my real test tomorrow and I don't understand the question. Can anyone help explain it plz any help would be great.
Step-by-step explanation:
Kaylee deposited $1,450 in an account that earns 2.596 interest compounded annually. Which function represents the situation, where tis
the time in years?
fit) = 1450(2.5)
f(t) = 1450(1.25)
FO) = 1450(.025)
f(t) = 1450(1,025)
Answer:
[tex]f(t) = 1450(1.025)^{t}[/tex]
Step-by-step explanation:
Given
[tex]P =1450[/tex] -- principal
[tex]r = 2.5\%[/tex] --- rate
[tex]n = 1[/tex] --- compounded once a year
Required
Determine the function for compound interest
Compound interest f(t) is calculated as:
[tex]f(t) =P(1 + r/n)^{nt[/tex]
So, we have:
[tex]f(t) = 1450(1 + 2.5\%/1)^{1 * t}[/tex]
[tex]f(t) = 1450(1 + 2.5\%)^{t}[/tex]
[tex]f(t) = 1450(1 + 0.025)^{t}[/tex]
[tex]f(t) = 1450(1.025)^{t}[/tex]
plsssssss help
got 20 mins
the question is: The sin of angle DCB is
Answer:
i. <DCB = [tex]53.13^{o}[/tex]
ii. Sin of <DCB = 0.8
Step-by-step explanation:
Let <DCB be represented by θ, so that;
Sin θ = [tex]\frac{opposite}{hypotenuse}[/tex]
Thus from the given diagram, we have;
Sin θ = [tex]\frac{4}{5}[/tex]
= 0.8
This implies that,
θ = [tex]Sin^{-1}[/tex] 0.8
= 53.1301
θ = [tex]53.13^{o}[/tex]
Therefore, <DCB = [tex]53.13^{o}[/tex].
So that,
Sin of <DCB = Sin [tex]53.13^{o}[/tex]
= 0.8
Sin of <DCB = 0.8
Please help, I can’t figure this answer out and I’m really struggling on it!
The exponent on the (x - 1) term include the following: A. 3.
What is an exponent?In Mathematics, an exponent is a mathematical operation that is commonly used in conjunction with an algebraic equation or expression, in order to raise a given quantity to the power of another.
Mathematically, an exponent can be represented or modeled by this mathematical expression;
bⁿ
Where:
the variables b and n are numbers (numerical values), letters, or an algebraic expression.n is known as a superscript or power.By critically observing the graph of this polynomial function, we can logically deduce that it has a zero of multiplicity 3 at x = 1, a zero of multiplicity 1 at x = 3, and zero of multiplicity 2 at x = 4;
x = 1 ⇒ x - 1 = 0.
(x - 1)³
x = 3 ⇒ x - 3 = 0.
(x - 3)
x = 4 ⇒ x - 4 = 0.
(x - 4)²
Therefore, the required polynomial function is given by;
P(x) = (x - 1)³(x - 3)(x - 4)²
Exponent of (x - 1)³ = 3.
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what is the equation for a Vertical Shift 5 units up?
Find the measure of C to the nearest tenth of a degree
An insurance company crashed four cars in succession at 5 miles per hour. The cost of repair for each of the four crashes was $421. 5452.5415, $232 Compute the range, sample variance, and sample standard deviation cost of repair, The range is 2-dollars? (Round to the nearest whole number as needed.) (Round to two decimal places as needed)
The range of the repair costs for the four car crashes is $5452.54 - $232 = $5220.54. The sample variance of the repair costs is $4,898,414.69, and the sample standard deviation is $2,214.17.
What are the range, sample variance, and sample standard deviation of the repair costs?The range of the repair costs for the four car crashes is the difference between the highest and lowest cost, resulting in a range of $5220.54. This indicates the variability in the repair costs. The sample variance is a measure of the average squared deviation from the mean, calculated to be $4,898,414.69. It shows the dispersion of the repair costs from the average. The sample standard deviation is the square root of the variance, amounting to $2,214.17. It provides a measure of how spread out the repair costs are, with a higher value indicating greater variability.
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Answer number two please please
Answer: 11
Step-by-step explanation:
Sooooo all I did was take the formula for the area of a triangle ( A= 1/2(b)(h) 0 and plug In the values. So 'b' would be 4 and 'h' would be 5.5. I assumed this rectangle had equal widths and equal heights.
a rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y = 2 − x 2 . what are the dimensions of such a rectangle with the greatest possible area?
To find the dimensions of the rectangle with the greatest possible area inscribed in the parabola y = 2 - x^2, we need to maximize the area function by determining the x-coordinate where the derivative of the area function is zero.
Let's consider a rectangle with its base on the x-axis, which means its height will be given by the y-coordinate of the parabola. The width of the rectangle will be twice the x-coordinate. Therefore, the area of the rectangle is given by A = 2x(2 - x^2).
To maximize the area, we take the derivative of A with respect to x and set it equal to zero to find critical points. Differentiating A, we get dA/dx = 4 - 6x^2.
Setting 4 - 6x^2 = 0 and solving for x, we find x = ±√(2/3).
Since the rectangle is inscribed, we consider the positive value of x. Therefore, the x-coordinate of the upper corner of the rectangle is √(2/3). Plugging this value back into the equation of the parabola, we get y = 2 - (√(2/3))^2 = 2 - 2/3 = 4/3.
Hence, the dimensions of the rectangle with the greatest possible area are a base of length 2√(2/3) on the x-axis and a height of 4/3.
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Assume x and y are functions of t.
Evaluate dy/dt for 4xy-3x+4y^3= -76 dx/dt =-8, x=4, and y=-2
The value of dy/dt for the given equation and values is -6.
To evaluate dy/dt, we can differentiate the given equation with respect to t using the chain rule. Starting with the equation 4xy - 3x + 4y^3 = -76, we differentiate both sides with respect to t.
Differentiating each term separately, we get:
(d/dt)(4xy) - (d/dt)(3x) + (d/dt)(4y^3) = 0
Using the chain rule, we can rewrite this as:
4(dy/dt)(x) + 4x(dy/dt) - 3(dx/dt) + 12y^2(dy/dt) = 0
Substituting the given values dx/dt = -8, x = 4, and y = -2, we have:
4(dy/dt)(4) + 4(4)(dy/dt) - 3(-8) + 12(-2)^2(dy/dt) = 0
Simplifying the equation, we get:
16(dy/dt) + 16(dy/dt) + 24 + 48(dy/dt) = 0
80(dy/dt) = -24
(dy/dt) = -24/80
(dy/dt) = -3/10
(dy/dt) = -0.3
Therefore, dy/dt evaluates to -0.3.
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