The compound statement "If there are penguins and the aquarium is full of fish, then this is an octopus" can be written in symbolic form as: r ∧ q → o
How to explain the informationLet's assign variables to represent the statements:
q: The aquarium is full of fish.
r: There are penguins.
The compound statement "If there are penguins and the aquarium is full of fish, then this is an octopus" can be written in symbolic form as: r ∧ q → o
∧ represents the logical operator "and" which connects the statements r and q.
→ represents the logical operator "implies" or "if...then".
o represents the statement "this is an octopus".
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The domain and ranger of a linear function is always all real numbers true or false ?
Answer:
Step-by-step explanation:
The domain and range of a linear function is always real numbers (T or F)
It is True. This is because of a couple of reasons.
1.) You cannot divide by 0.
2. A negative number cannot have its square root taken.
The range is determined by the domain in a linear function, and thus it must always consist of real numbers.
If a=-2 and b=-4 find the value of 3a+b+4
Answer:
-6
Step-by-step explanation:
hey laudu
\(3a + b + 4\)
\(3( - 2) + ( - 4) + 4\)
\( - 6 - 4 + 4\)
\( - 6\)
Step-by-step explanation:
❀ \( \underline{ \underline{ \text{Given}}} : \)
a = -2 & b = - 4❀ \( \underline{ \underline{ \text{To \: find}}} : \)
value of 3a + b + 4✏ \( \underline{ \underline{ \text{Solution}}} : \)
All you need to do is plug the value and simplify !
⟶ \( \tt{3 \times ( - 2) + ( - 4) + 4}\)
⟶ \( \tt{ - 6 + ( - 4) + 4}\)
⟶ \( \tt{ - 6 - 4 + 4}\)
⟶ \( \tt{ - 10 + 4}\)
⟶ \( \boxed{ \tt{ - 6}}\)
\( \red{ \boxed{ \boxed{ \tt{Our \: final \: answer : \boxed{ \underline{ - 6}}}}}}\)
Hope I helped ! ♪
Have a wonderful day / night ! ♡
▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁
Find f. (Use C for the constant of the first antiderivative and D for the constant of the second antiderivative.)
f ''(x) = 2x + 4ex
After integration, the required function f is (2x³ - sin (x) + Cx + D).
What is the integration of 'xⁿ' and 'sin (x)'?
\(\int {x^{n} } \, dx = \frac{x^{n+1} }{n+1} + C\\\\\\\int {sinx} \, dx = -cosx + C\)
Given, f''(x) = 12x + sin x
Therefore,
\(\int {f''(x)} \, dx \\\\=\int {f'(x)} \, dx \\\\\\= \int{12x + sin x} \, dx + C\\\\= 6x^{2} - cosx + C\\\)
Again, f'(x) = 6x - cos (x) + C
Therefore,
\(\int {f'(x)} \, dx\\ \\=\int {f(x)} \, dx \\\\= \int {6x^{2} - cosx + C } \, dx \\\\= 2x^{3} - sinx + Cx + D\)
Therefore, the required function is (2x³ - sin (x) + Cx + D).
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The sum of two numbers is 79,and their difference is 33
Answer: infinite answers, jk there is like a lot tho
Step-by-step explanation: The sum of x and y is 79. In other words, x plus y equals 79 and can be written as equation A:
x + y = 79
The difference between x and y is 23. In other words, x minus y equals 23 and can be written as equation B:
x - y = 23
Now solve equation B for x to get the revised equation B:
x - y = 23
x = 23 + y
Then substitute x in equation A from the revised equation B and then solve for y:
x + y = 79
23 + y + y = 79
23 + 2y = 79
2y = 56
y = 28
Now we know y is 28. Which means that we can substitute y for 28 in equation A and solve for x:
x + y = 79
x + 28 = 79
X = 51
Summary: The sum of the two numbers is 79 and their difference is 23. What are the two numbers? Answer: 51 and 28 as proven here:
Sum: 51 + 28 = 79
Difference: 51 - 28 = 23
Please give me brainliest
Reuben made a shirt using 7/8yards of red fabric and 1/4yards of yellow fabric. How many more yards of red fabric did Reuben use?
Answer and Step-by-step explanation:
To find out how many more yards of red fabric Reuben used, we need to subtract the amount of yellow fabric from the amount of red fabric. Since the two fractions have different denominators, we need to find a common denominator before subtracting them. The least common multiple of 8 and 4 is 8, so we can rewrite both fractions with a denominator of 8:
7/8 - 1/4 = 7/8 - (1/4) * (2/2) = 7/8 - 2/8 = (7 - 2)/8 = 5/8
So, Reuben used 5/8 yards more red fabric than yellow fabric.
Solve for X to the nearest tenth.
Answer:
x ≈ 24.4
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to RightEquality Properties
Multiplication Property of Equality Division Property of Equality Addition Property of Equality Subtraction Property of EqualityTrigonometry
[Right Triangles Only] SOHCAHTOA[Right Triangles Only] cosθ = adjacent over hypotenuseStep-by-step explanation:
Step 1: Define
Identify variables
Angle θ = 38°
Adjacent Leg = x
Hypotenuse = 31 cm
Step 2: Solve for x
Substitute in variables [cosine]: cos38° = x/31[Multiplication Property of Equality] Multiply 31 on both sides: 31cos38° = xRewrite: x = 31cos38°Evaluate: x = 24.4283Round: x ≈ 24.4The ratio of moose to reindeer is 5 to 4. If
there are 60 moose, how many reindeer are
there?
Answer:
48 reindeer
Step-by-step explanation:
the 5 part of the ratio relates to 60 moose , then
60 ÷ 5 = 12 ← value of 1 part of the ratio
then
4 × 12 = 48 ← number of reindeer
P(z > c) = 0.8914
Find C
The value of c is given as follows:
c = -1.235.
How to obtain probabilities using the normal distribution?The z-score of a measure X of a variable that has mean symbolized by \(\mu\) and standard deviation symbolized by \(\sigma\) is obtained by the rule presented as follows:
\(Z = \frac{X - \mu}{\sigma}\)
The z-score represents how many standard deviations the measure X is above or below the mean of the distribution, depending if the obtained z-score is positive or negative.Using the z-score table, the p-value associated with the calculated z-score is found, and it represents the percentile of the measure X in the distribution.For this problem, we have that P(z > c) = 0.8914, meaning that Z when X = c has a p-value of 1 - 0.8914 = 0.1086, hence the value of c is given as follows:
c = -1.235.
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2. Find the surface area of the sphere below. Leave your answer in terms of . Type 'pi' to represent Pi
Given the surface area of the sphere
Formular
\(\begin{gathered} \text{Surface Area = 4}\pi r^2 \\ S\mathrm{}A\text{ = 4 }\pi(2.5)^2 \\ S.A\text{ = 25 }\pi cm^2 \end{gathered}\)Hence the surface area of the sphere = 25 π cm²
The odds against an event are 19 to 12 find the probability that The event will occur
Answer:
the event is odds against an event of the 19 to 12 is probability is the 19
The probability of occurring the event is 12/31.
What is probability?Probability is the chance of occurrence of a certain event out of the total no. of events that can occur in a given context.
We know if the odds against an event is a : b then the probability of occurring the event is b/(a + b).
Given, The odds against an event are 19 to 12.
∴ The probability of occurring the event is 12/(19 + 12) = 12/31.
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Point A is located at (1, 5), and point M is located at (-1, 6). If point M is the midpoint of AB, find the location of point B.
The required coordinates of point B are (-3, 7) where point M is the midpoint of AB.
What is Midpoint?A midpoint is defined as in the middle of the line connecting two points a position known as a midpoint. A location in the middle of a line connecting the two points that are equally far from both points is the midpoint.
Point A is located at (1, 5), and point M is located at (-1, 6). If point M is the midpoint of AB
We can use these coordinates to find the coordinates of point B, which is the midpoint of line segment AB.
Let the coordinates of B would be (x, y)
Substituting the coordinates of points A into the midpoint formula gives us :
-1 = (x+1)/2 ; 6 = (y+5)/2
-2 = x +1 ; 12 = y + 5
x = -3; y = 7
Therefore, the required coordinates of point B are (-3, 7).
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3x+4
12. Simplify +
x+2
x²+2x
2x+4
Answer:
\(\dfrac{x^2+8x+8}{2x+4}\\\\\)
Step-by-step explanation:
Simplify:
\(\dfrac{3x + 4}{x +2}+\dfrac{x^2+2x}{2x+ 4}\)
Multiply the denominator and the numerator of the first term by 2,
\(=\dfrac{2*(3x +4)}{2*(x+ 2)}+\dfrac{x^2+2x}{2x+4}\\\\\\=\dfrac{2*3x+2*8}{2*x +2*2}+\dfrac{x^2+2x}{2x+4}\\\\\\=\dfrac{6x+8}{2x+4}+\dfrac{x^2+2x}{2x+4}\)
\(\sf = \dfrac{6x+8+x^2+2x}{2x+4}\\\\\text{\bf Combine like terms,}\\\\=\dfrac{x^2+8x+8}{2x+4}\\\\\)
There are six pizzas there are 36 players on the team. If each member has one slice and each pizza has eight slices how much pizza in fraction form will be left over.
There will be 12 slices of pizza left over, which can be expressed as 3/2 or 1 1/2 slices in fraction form.
To find out how much pizza will be left over, we need to calculate the total number of slices available and subtract the number of slices consumed by the players.
Given that there are six pizzas and each pizza has eight slices, the total number of slices available is 6 pizzas * 8 slices/pizza = 48 slices.
Since there are 36 players on the team, and each player has one slice, the total number of slices consumed is 36 slices.
To determine the remaining slices, we subtract the consumed slices from the total available slices: 48 slices - 36 slices = 12 slices.
Therefore, there will be 12 slices of pizza left over.
To express this as a fraction, we can write it as 12/1, which simplifies to 12. This means there will be 12 whole slices of pizza left over.
If you would like to express it as a fraction in its simplest form, you can write it as 12/8. By dividing both the numerator and denominator by their greatest common divisor, which is 4, we get 3/2. So, in fraction form, there will be 3/2 or 1 1/2 slices of pizza left over.
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Adding and subtracting rational expression
What is the sun if the rational expressions
2x+3/3X + x/x+1
A. 2x^2+3
3x^2+3x
B. 3x+3
4x+1
C. 5x^2+5x+3
3x^2+3x
D. 3x^2+2x+4
4x+1
Answer:
C
Step-by-step explanation:
Use cross multiplication method to find the answer:
\(\frac{2x+3}{3x} +\frac{x}{x+1} = \frac{(2x^2+5x+3) + (3x^2)}{2x^2+3x} = \frac{5x^2+5x+3}{3x^2+3x} \)
Sample Responses Label axes according to input
and output variables. Plot the ordered pair of the
independent and dependent variable on the
coordinate plane. Identify if there is a relationship
Which of the following did you include in your
response?
O Label the x-axis the input variable, age.
O Label the y-axis the output variable, texting speed.
O Plot the points according to age and texting speed.
O Identify a relationship between the change in
texting speed as age increases.
The following elements were included in the response: labeling x-axis as age, labeling y-axis as texting speed, plotting points according to age and texting speed, and identifying a relationship between texting speed and age.
In the response, the following elements were included:
1. Labeling the x-axis as the input variable, age: This is important to indicate the independent variable being plotted.
2. Labeling the y-axis as the output variable, texting speed: This is crucial to indicate the dependent variable being represented.
3 Plotting the points according to age and texting speed: This involves placing the ordered pairs (age, texting speed) on the coordinate plane, with age values on the x-axis and texting speed values on the y-axis.
4. Identifying a relationship between the change in texting speed as age increases: By analyzing the plotted points and examining the pattern or trend, one can determine if there is a relationship between age and texting speed. For example, if the texting speed generally increases as age increases or if there is a linear or nonlinear relationship between the two variables.
Including all of these elements allows for a comprehensive analysis of the relationship between age and texting speed, providing a visual representation and enabling the identification of any patterns or trends.
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The residual plot shows the residuals for the day of the month and the amount in Kali’s checking account. Which statement best assesses the linearity of the relationship between the day of the month and account balance if the scatterplot appears to be reasonably linear?
A) Because the residual plot has an obvious pattern, and the scatterplot appears linear, it is appropriate to use the line of best fit to predict the account balance based on the day of the month.
B) Because the residual plot has an obvious pattern, and the scatterplot appears linear, it is not appropriate to use the line of best fit to predict the account balance based on the day of the month.
C) Because the residual plot has no obvious pattern, and the scatterplot appears linear, it is appropriate to use the line of best fit to predict the account balance based on the day of the month.
D) Because the residual plot has no obvious pattern, and the scatterplot appears linear, it is not appropriate to use the line of best fit to predict the account balance based on the day of the month.
The best assessment of the linearity of the relationship between the day of the month and account balance would be "Because the residual plot has no obvious pattern, and the scatterplot appears linear, it is appropriate to use the line of best fit to predict the account balance based on the day of the month."The correct answer is option C.
When assessing linearity, it is important to examine both the scatterplot and the residual plot. The scatterplot is used to visualize the relationship between the variables, while the residual plot helps us assess the appropriateness of a linear model by examining the pattern of the residuals (the differences between observed and predicted values).
If the scatterplot appears reasonably linear, it suggests that there is a linear relationship between the variables. In this case, since the scatterplot appears linear, it supports the use of a linear model to predict the account balance based on the day of the month.
Furthermore, the residual plot is used to check for any patterns or systematic deviations from the line of best fit. If the residual plot exhibits no obvious pattern and the residuals appear randomly distributed around zero, it indicates that the linear model captures the relationship adequately.
Therefore, if the residual plot shows no obvious pattern, it provides further evidence in favor of using the line of best fit to predict the account balance based on the day of the month.
In conclusion, when the scatterplot appears linear and the residual plot shows no obvious pattern, it is appropriate to use the line of best fit to predict the account balance based on the day of the month.
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Answer:
person above
Step-by-step explanation:
the obvious
Construct an exponential equation for the data presented in the table.
A) y = 1 * (x)^2
B) y = 2 * (x)^1
C) y = 1 * (2)^x
D) y = 2 * (1)^x
Answer:
C
Step-by-step explanation:
100 Points! Geometry question. Photo attached. Find the measure. Please show as much work as possible. Thank you!
Answer:
The answer would lie within 31 degrees of MP and also as in PM.
Answer:
central m arc MP=118°
Step-by-step explanation:
here
central m arc MN=2* inscribed m arc MN=2*31=62°
again
central m arc MN+ central m arc MP=180° being linear pair
substituting value
62°+central m arc MP=180°
central m arc MP=180°-62°
central m arc MP=118°
Equipment that costs $12,000 and has accumulated depreciation of $9,000 sold for $2,600. The journal entry would include a
Multiple Choice
The journal entry for Equipment that costs $12,000 and has accumulated depreciation of $9,000 sold for $2,600 would include a C) debit to Loss on Sale of Equipment for $400.
What is the journal entry?The journal entry refers to the initial record of a business transaction.
When an item of equipment is sold, the journal entries include a credit to the equipment account for the cost, a debit to the accumulated depreciation account and a debit or credit to the Sale of Equipment account.
Equipment cost = $12,000
Accumulated depreciation = $9,000
Book value of equipment = $3,000
Sales proceeds = $2,600
Loss on Sale of Equipment = $400 ($3,000 - $2,600)
Thus, the loss on sale of equipment is recorded because the book value is more than the sales proceeds.
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Question Completion:A) credit to Accumulated Depreciation for $9,000
B) debit to Loss on Sale of Equipment for $9,400.
C) debit to Loss on Sale of Equipment for $400
D) credit to Equipment for $3,000
The grades on a physics midterm at Covington are roughly symmetric with = 72 and = 2.0. Stephanie scored 74 on the exam. Find the scores for Stephanie’s exam grade. Round to two decimal places
Answer:1.00
Step-by-step explanation:
A=74-72/2.0
And I did it on kahn
The z - score for the Stephanie’s exam grade will be 1.
What is Standard deviation?
Standard deviation is the measure of dispersed the data is in relation to the mean. Low standard deviation means data are clustered around the mean, and high standard deviation indicates data are more spread out.
Given that;
The grades on a physics midterm at Covington are roughly symmetric with μ = 72 and σ = 2.0.
And, Stephanie scored 74 on the exam.
Now,
The formula for the z - score = ( X - μ ) / σ
Substitute all the values, we get;
The z - score for the Stephanie’s exam grade = ( 74 - 72 ) / 2.0
= 2 / 2
= 1
Thus, The z - score for the Stephanie’s exam grade will be 1.
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If AABC= AMNO, then what corresponding angle is congruent to angle N?
Answer:
.aabc=amn0#1211
Step-by-step explanation:
nosascoOsasco n=12.8
Question 3 (1 point)
Karl wants to find the width RQ of a river. He starts at point R, and walks
perpendicular along the edge of the river 42 ft and marks point S. He then walks 28
ft further and marks point T. He turns 90° and walks until his location (point U), point
S, and point Q are collinear. Suppose TU= 68 ft. What is the width of the river in
feet?
The width RQ of the river is approximately 61.98 ft.
To find the width RQ of the river, we can use the properties of perpendicular lines and collinearity.
Given that Karl starts at point R and walks perpendicular along the edge of the river 42 ft to point S, we can draw a line segment RS of length 42 ft.
From point S, Karl walks 28 ft further to point T. We can draw another line segment ST of length 28 ft.
Now, Karl turns 90° from point T and walks until his location (point U), point S, and point Q are collinear. Let's denote the length of this line segment as UQ.
From the given information, we know that TU = 68 ft.
Since U, S, and Q are collinear, we can form a right triangle by connecting UQ and US.
The length of UQ can be found using the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
In this case, we have:
UQ² = TU² - US²
UQ² = 68² - 28²
UQ² = 4624 - 784
UQ² = 3840
Taking the square root of both sides, we have:
UQ = √3840
UQ ≈ 61.98 ft
Therefore, the width RQ of the river is approximately 61.98 ft.
Note: It's important to keep in mind that this solution assumes the river is a straight line and that Karl's path is perpendicular to the river's edge. In reality, the river's edge may not be perfectly straight, and the path Karl walks may not be exactly perpendicular.
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Verify that the following function is a probability mass function, and determine the requested probabilities.
fx = (8/7(1/2)x
x = 0, 1, 2, ...
A. P(X = 2).
B. P(X ≤ 2).
C. P(X > 2).
D. P(X ≥ 1).
Is the function a probability mass function?
Answer:
yes
Step-by-step explanation:
Maxwell buys a bag of dog food that contains 18 cups of food, Maxwell feeds his dog 1 cups each day. Maxwell says he will be able to feed his dog for 27 days.
correct?
Answer:
Incorrect
Step-by-step explanation:
Well since there is only 18 cups of dog food he will only be able to feed his dog for 18 days since he feeds his dog 1 cup a day.
Answer:
Incorrect
Explanation:
Maxwell's dog eats one cup of food a day. The bag contains only eighteen cups, so it will feed the dog for eighteen days. There will not be enough food to last Maxwell for 27 days. So, the answer is incorrect.
The equation y=195(1.10)x can be used to represent growth in the number of Salad Supreme restaurants since 2005. Which is the most reasonable conclusion based on this information? A. Salad Supreme had 110 restaurants in 2005. B. Salad Supreme is increasing the number of restaurants by 10% each year. C. Salad Supreme is increasing the number of restaurants by 195 each year. D. Salad Supreme is increasing the number of restaurants by 110% each year.
The most reasonable conclusion based on the given information is: Salad Supreme is increasing the number of restaurants by 10% each year. Option B
How to determine the most reasonable conclusionBased on the given equation y = 195(1.10)^x, where x represents the number of years since 2005 and y represents the number of Salad Supreme restaurants, we can draw the following conclusions:
A. Salad Supreme had 110 restaurants in 2005: This conclusion is not supported by the given equation.
B. Salad Supreme is increasing the number of restaurants by 10% each year: This conclusion is supported by the equation.
C. Salad Supreme is increasing the number of restaurants by 195 each year: This conclusion is not supported by the equation.
D. Salad Supreme is increasing the number of restaurants by 110% each year: This conclusion is not supported by the equation. The growth factor of 1.10 (or 10%) indicates a 10% increase each year, not 110%.
Therefore, the most reasonable conclusion based on the given information is: Salad Supreme is increasing the number of restaurants by 10% each year.
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Here is a shape.
Write down the mathematical name for the shape.
No spam links, give the answer.
Answer:
hexagon is the six sided mathematical shape
Answer:
hexagon
Step-by-step explanation:
because it is having a six sided figure
Given theta equals 675° convert the value of data to radians and find Secant of theta
Answer
• θ = 15/4π ≈ 11.78rad
\(sec(\theta)=\sqrt{2}\)Explanation
Given:
• θ = 675°
,• Convert to radians
,• Find secant (θ)
Procedure
To convert θ to radians we have to remember how many radians is xº:
\(\frac{2\pi rad}{180\degree}\)Then, we multiply this conversion times the degrees given:
\(\theta=675\degree\cdot\frac{\pi rad}{180\degree}=\frac{15}{4}\pi\text{ rad}\approx11.78\text{ rad}\)Finally, finding the secant of θ:
\(\text{ sec\lparen}\theta)=\text{ sec\lparen}11.78)=\sqrt{2}\)Find y' for the following.
Answer:
\(\displaystyle y' = \frac{\sqrt{y} + y}{4\sqrt{x}y \Big( y^\Big{\frac{3}{2}} + \sqrt{y} + 2y \Big) + \sqrt{x} + x}\)
General Formulas and Concepts:
Calculus
Differentiation
DerivativesDerivative NotationDerivative Property [Multiplied Constant]: \(\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)\)
Derivative Property [Addition/Subtraction]: \(\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]\)
Basic Power Rule:
f(x) = cxⁿf’(x) = c·nxⁿ⁻¹Derivative Rule [Quotient Rule]: \(\displaystyle \frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}\)
Derivative Rule [Chain Rule]: \(\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)\)
Implicit Differentiation
Step-by-step explanation:
Step 1: Define
Identify
\(\displaystyle \frac{\sqrt{x} + 1}{\sqrt{y} + 1} = y^2\)
Step 2: Differentiate
Implicit Differentiation: \(\displaystyle \frac{dy}{dx} \bigg[ \frac{\sqrt{x} + 1}{\sqrt{y} + 1} \bigg] = \frac{dy}{dx}[ y^2]\)Quotient Rule: \(\displaystyle \frac{(\sqrt{x} + 1)'(\sqrt{y} + 1) - (\sqrt{y} + 1)'(\sqrt{x} + 1)}{(\sqrt{y} + 1)^2} = \frac{dy}{dx}[ y^2]\)Rewrite: \(\displaystyle \frac{(x^\Big{\frac{1}{2}} + 1)'(y^\Big{\frac{1}{2}} + 1) - (y^\Big{\frac{1}{2}} + 1)'(x^\Big{\frac{1}{2}} + 1)}{(y^\Big{\frac{1}{2}} + 1)^2} = \frac{dy}{dx}[ y^2]\)Basic Power Rule [Addition/Subtraction, Chain Rule]: \(\displaystyle \frac{\frac{1}{2}x^\Big{\frac{-1}{2}}(y^\Big{\frac{1}{2}} + 1) - \frac{1}{2}y^\Big{\frac{-1}{2}}y'(x^\Big{\frac{1}{2}} + 1)}{(y^\Big{\frac{1}{2}} + 1)^2} = 2yy'\)Factor: \(\displaystyle \frac{\frac{1}{2} \bigg[ x^\Big{\frac{-1}{2}}(y^\Big{\frac{1}{2}} + 1) - y^\Big{\frac{-1}{2}}y'(x^\Big{\frac{1}{2}} + 1) \bigg] }{(y^\Big{\frac{1}{2}} + 1)^2} = 2yy'\)Rewrite: \(\displaystyle \frac{x^\Big{\frac{-1}{2}}(y^\Big{\frac{1}{2}} + 1) - y^\Big{\frac{-1}{2}}y'(x^\Big{\frac{1}{2}} + 1)}{2(y^\Big{\frac{1}{2}} + 1)^2} = 2yy'\)Rewrite: \(\displaystyle x^\Big{\frac{-1}{2}}(y^\Big{\frac{1}{2}} + 1) - y^\Big{\frac{-1}{2}}y'(x^\Big{\frac{1}{2}} + 1)}= 4yy'(y^\Big{\frac{1}{2}} + 1)^2\)Isolate y' terms: \(\displaystyle x^\Big{\frac{-1}{2}}(y^\Big{\frac{1}{2}} + 1) = 4yy'(y^\Big{\frac{1}{2}} + 1)^2 + y^\Big{\frac{-1}{2}}y'(x^\Big{\frac{1}{2}} + 1)}\)Factor: \(\displaystyle x^\Big{\frac{-1}{2}}(y^\Big{\frac{1}{2}} + 1) = y' \bigg[ 4y(y^\Big{\frac{1}{2}} + 1)^2 + y^\Big{\frac{-1}{2}}(x^\Big{\frac{1}{2}} + 1)} \bigg]\)Isolate y': \(\displaystyle \frac{x^\Big{\frac{-1}{2}}(y^\Big{\frac{1}{2}} + 1)}{4y(y^\Big{\frac{1}{2}} + 1)^2 + y^\Big{\frac{-1}{2}}(x^\Big{\frac{1}{2}} + 1)} = y'\)Rewrite/Simplify: \(\displaystyle y' = \frac{\sqrt{y} + y}{4\sqrt{x}y \Big( y^\Big{\frac{3}{2}} + \sqrt{y} + 2y \Big) + \sqrt{x} + x}\)Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
Book: College Calculus 10e
The mean height of three boys is 150 centimeters. The mean height of two girls is 145.Find the mean height of five students
A mean is an arithmetic average of a set of observations. The mean height of the five students is 148.
What is Mean?A mean is an arithmetic average of a set of observations. it is given by the formula,
Mean = (Sum of observations)/Number of observations
Given that the mean height of three boys is 150 centimeters. Therefore, the sum of height of the three boys is,
Mean Height of boys = Sum of height of boys / Number of boys
150 = Sum of height of boys/ 3
Sum of height of boys = 150 x 3 = 450
Also, given that the mean height of two girls is 145 centimeters. Therefore, the sum of height of the two girls is,
Mean Height of girl = Sum of height of girls/ Number of girls
145 = Sum of height of girls/ 2
Sum of height of girls = 145 x 2 = 290
Further, the mean height of five students is,
Mean height of five students = Sum of height of students / Number of students
Mean height of five students = (450 + 290)/ 5
Mean height of five students = 740 / 5
= 148
Hence, the mean height of the five students is 148.
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mariana and her children went into a movie theater that sells bags of popcorn for $6 each and candies for $4 each. mariana has $80 to spend and must buy at least 16 bags of popcorn and candies altogether. if a represents the number of bags of popcorn purchased and y represents the number of candies purchased, write and solve a system of inequalities graphically and determine one possible solution.
One possible solution from the graph(attached at the end of the solution) is selling 9 bags of popcorn and 5 bags of candies.
As per the given data:
The number of popcorn bags is x
The number of candies is y
The cost of each popcorn bag is $7
The cost of each candy is $4
She has $80 to spend on them
Inequality is the nonequal comparison of two or more variables and numbers.
So the first inequality is
7 x + 4 y ≤ 80
Since she must buy no less than 16 bags of popcorn and candies
The second inequality is
x + y ≥ 16
The red area represents the 1st inequality
The blue area represents the 2nd inequality
The solutions lie in the common part of red and blue colors
One possible solution is ordered pair (9,5)
(Graph attached at the end of solution)
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