Answers
From the Venn diagram, the sum of the numbers in all the circles and overlapping areas should add up to the total number of students surveyed, which is 85.
What is the Venn diagramIn the Venn diagram:
M represents the set of students who like mango.B represents the set of students who like banana.K represents the set of students who like kiwi fruit.The overlapping areas represent students who like two or three flavors.The "X" in the center represents the 2 students who like all three flavors.Using the information given:
35 students liked mango, so the circle representing mango should contain 35.37 students liked banana, so the circle representing banana should contain 37.26 students liked kiwi fruit, so the circle representing kiwi fruit should contain 26.2 students liked all three flavors, so the overlapping area of all three circles should contain 2.20 liked both mango and banana, so the overlapping area of mango and banana should contain 20.14 liked mango and kiwi fruit, so the overlapping area of mango and kiwi fruit should contain 14.3 liked banana and kiwi fruit, so the overlapping area of banana and kiwi fruit should contain 3.Note that the sum of the numbers in all the circles and overlapping areas should add up to the total number of students surveyed, which is 85.
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Related Questions
Use implicit differentiation to find an equation of the tangent line to the curve at the given point.
x2+y2=(2x2+2y2−x)2,(0,1/2)
Answers
As a result, y = -x + 1/2 is the equatiοn οf the tangent line tο the curve x² + y² = (2x² + 2y² - x)² at the pοsitiοn (0,1/2).
What is equatiοn?An equatiοn in mathematics is a claim that twο mathematical expressiοns are equivalent. An equatiοn has an equal sign, typically has οne οr mοre factοrs, and can be sοlved fοr by applying algebraic οr numerical techniques. Equatiοns are used tο represent real-wοrld phenοmena, resοlve issues, and make predictiοns in many branches οf mathematics, science, and engineering.
given:
We must first use implicit differentiatiοn tο determine the derivative οf the curve in οrder tο determine the equatiοn οf the tangent line tο the curve at the pοsitiοn (0,1/2).
By taking the derivative οf the equatiοn's twο sides with regard tο x, we arrive at:
dy/dx = 2x + 2y
(2x² + 2y²- x) * (4x - 1) (4x - 1)
When we sοlve fοr dy/dx and simplify this equatiοn, we get:
[(1 - 4x)(2x + 2y)] Equals dy/dx / [4(2x² + 2y² - x) - 2y(1 - 4x)]
The slοpe οf the tangent line at the pοint (0,1/2) can nοw be calculated by plugging in the numbers x = 0 and y = 1/2:
dy/dx = [(1 - 4(0))
(2(0) + 2(1/2))] / [4(2(0)² + 2(1/2)² - 0) - 2(1/2)(1 - 4(0))] = -1
Therefοre, the tangent line's slοpe at the pοsitiοn (0,1/2) is negative οne. We can use the pοint-slοpe fοrm οf a line tο determine the equatiοn οf the tangent line:
where (x1,y1) is the tangency pοint and m is the inclinatiοn οf the tangent line. When we enter the numbers we have, we οbtain:
y - 1/2 = -1(x - 0) (x - 0)
When we simplify this sοlutiοn, we οbtain:
y = -x + 1/2
As a result, y = -x + 1/2 is the equatiοn οf the tangent line tο the curve x² + y2 = (2x² + 2y² - x)² at the pοsitiοn (0,1/2).
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A law firm billed a client $3,300 for work performed in the current month. Which of the following general journal entries will the firm make to record this transaction?
Answers
Therefore , the solution of the given problem of unitary method comes out to be amount billed to the client.
An unitary method is what?To finish the job using the unitary technique, the equation from this nano section should be multiplied by two. In essence, when a wanted object is present, both the characterised by either a group and the pigment sections are omitted from the unit method. A variable fee of Inr ($1.01), for instance, would be paid for 40 pens. It's conceivable that one nation will have complete control over the strategy used to achieve this. Almost all living things have a unique trait.
Here,
$3,300 is debited from Accounts Payable - Client's Name.
Credit: $3,300 in revenue
This shows the revenue earned by the law company for the job done and records the increase in accounts receivable for the amount billed to the client.
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Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f. (If an answer does not exist, enter DNE.) f(t) = 5 cos t, −3/2 ≤ t ≤ 3/2
Answers
Therefore , the solution of the given problem of graph comes out to be the graph is a symmetric wave that oscillates between 5 and 5.
A graph is what?Mathematicians use graphs to logically chart or illustrate claims rather than values in an interesting visual way. Usually, a graph point shows the connection between two or more objects. A graph is a particular kind of non-linear railway assembly composed of nodes and lines. The nodes, also known as the borders, should be joined together with glue. The node numbers in this network included 1, 2, 3, and 5, while the edge numbers included 1, 2, 3, and 4, along with (2.5), (3.5), (4.5), but also (4.5). (4.5).
Here,
Plotting a few important spots will help us later when we sketch the graph of f(t) = 5 cos(t).
Since the function has a phase of 2, we can concentrate on the range from 3/2 to 3/2. We possess
=> f(−3/2) = 5 cos(−3/2) ≈ 4.515
=> f(−π) = 5 cos(−π) = −5
=> f(−1/2) = 5 cos(−1/2) ≈ 4.045
=> f(0) = 5 cos(0) = 5
=> f(1/2) = 5 cos(1/2) ≈ 4.045
=> f(π) = 5 cos(π) = −5
=> f(3/2) = 5 cos(3/2) ≈ 4.515
Using these coordinates, we can create the following f(t) graph:
With maximum positions at t = /2 and t = /2, and minimum points at t = 3/2, t = 1/2, t = 1/2, and t = 3/2,
the graph is a symmetric wave that oscillates between 5 and 5.
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What is the total cost of a $713 tablet computer that is on sale at 11% off if the local sales tax rate is 7%?
The total cost of the tablet is S
(Round to two decimal places as needed.)
Answers
The total cost of the tablet is $684.48
How to calculate the total cost of the tablet?The first step is to write out the parameters given in the question
The cost of the tablet computer is $713
The tablet computer is on sale at 11%
Next is to multiply the total cost by the sales percent
11/100 × 713
= 0.11 × 713
= 78.43
= 713 - 78.43
= 634.57
The local sales tax rate is 7%
= 7/100 × 713
= 0.07 × 713
= 49.91
The total cost can be calculated as follows
= 49.91 + 634.57
= 684.48
Hence the total cost of the tablet is $684.48
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Every year Marla collects
acorns. If she collects 1,378
acorns each year for 7
years, how many acorns will
she have?
Answers
Answer:
9,646
Step-by-step explanation:
1378 acorns x 7 years = 9,646 acorns
what is the value of the x-coordinate of point A
Answers
The value of the x-coordinate of point A is -0.342
What is the value of the x-coordinate?If we measure the angle from the x-axis, we will get:
angle = 90° + 20° = 110°
Now remember that the rectangular coordinates are for a point with radius R and defined by an angle a are given by the expressions below:
x = R*cos(a)
y =R*sin(a)
Where in this case we can see that we have an unit circel, thus, R = 1, and a is the angle, in this case 110° as we saw above.
Then the x-coordinate of point A is:
x = 1*cos(110°) = -0.342
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What is the rate (i.e. Slope) between these two points? (-5,0) and (0, 6) *
Answers
slope = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are the coordinates of the two points.
Using the given points (-5,0) and (0,6), we can plug in the values and get:
slope = (6 - 0) / (0 - (-5)) = 6 / 5
Therefore, the rate (i.e. slope) between the two points is 6/5, which means that for every 5 units to the right, there is a 6-unit increase in the vertical direction. Alternatively, for every 5 units to the left, there is a 6-unit decrease in the vertical direction.
39.6% of consumers believe that cash will be obsolete in the next 20 years. Assume that 6 consumers are randomly selected. Find the probability that fewer than 3 of the selected consumers believe that cash will be obsolete in the next 20 years.
Answers
The probability that fewer than 3 of the selected consumers believe that cash will be obsolete in the next 20 years IS 0.528.
The binomial probability:The binomial probability formula is used to calculate the probability of getting exactly k successes in n independent trials, where each trial has only two possible outcomes (success or failure) and the probability of success is denoted as p. The formula is:
[tex]P(X=k) = C(n,k)\times p^k\times(1-p)^{(n-k)[/tex]Here we have
39.6% of consumers believe that cash will be obsolete in the next 20 years.
Let p be the probability that a consumer believes that cash will be obsolete
=> p = 39.6% or 0.396.
Then, the probability that a consumer does not believe that cash will be obsolete is q = 1 - p = 0.604.
We need to find the probability that less than 3 of the selected consumers believe that cash will be obsolete.
P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)
Using the binomial probability formula, we have:
[tex]P(X = k) = C(n,k) \times p^k\times q^{(n-k)[/tex]
=> P(X = 0) = C(6, 0) × (0.396)⁰ × (0.604)⁶ = 0.048
=> P(X = 1) = C(6, 1) × (0.396)¹ × (0.604)⁵ = 0.18
=> P(X = 1) = C(6, 2) × (0.396)² × (0.604)⁴ = 0.3
P(X < 3) = 0.048 + 0.18 + 0.3 = 0.528
Therefore,
The probability that fewer than 3 of the selected consumers believe that cash will be obsolete in the next 20 years IS 0.528.
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the ___ appears on correspondence, such as invoices and statements.
Answers
The blank in the sentence can be filled by the word "date." The date appears on correspondence, such as invoices and statements, to indicate the day on which the document was issued or created.
what is expression ?Multiplying, dividing, adding, and subtracting are all mathematical operations. As an example, consider the following expression: Expression, mathematics, and a numeric value Numbers, parameters, and functions are the components of an expression in mathematics. Using opposing words and phrases is possible. An expression, sometimes referred to as an algebraic expression, is any mathematical statement that includes variables, numbers, and a mathematical action between them. For instance, the expression 4m + 5 is made up of the expressions 4m and 5, as well as the variable m from the previous equation, all of which are separated by the mathematical symbol +.
The blank in the sentence can be filled by the word "date." The date appears on correspondence, such as invoices and statements, to indicate the day on which the document was issued or created.
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EASY MATH POINTS!
Answer from the screenshot below :)
Answers
The domain and range for the mapping diagram are given as follows:
Domain: {-9, -0.5, 1, 2.1, 6}.Range: {-7, 1, 6, 20, 154}.The diagram represents a function, as each value of the input is mapped to a single value of the output.
How to obtain the domain and the range of the diagram?The concepts for the domain and the range of the diagram are defined as follows:
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Jack has 7 yards of rope. He wants to cut it into pieces of different sizes. Jacks needs 84 inches of rope to tie some packages and 4 feet of rope for another project. Does Jack have enough rope? Explain.
Answers
Jack has enough rope for both projects.
What is multiplication?Multiplication is a type of mathematical operation. The repetition of the same expression types is another aspect of the practice.
For instance, the expression 2 x 3 indicates that 3 has been multiplied by two.
Given:
Jack has 7 yards of rope.
He wants to cut it into pieces of different sizes.
Jack needs 84 inches of rope to tie some packages and 4 feet of rope for another project.
First, we need to convert all the measurements to the same units.
Let's convert yards to inches and feet to inches:
7 yards = 21 feet = 252 inches
4 feet = 48 inches
Now, let's add up the lengths of rope that Jack needs:
84 inches + 48 inches = 132 inches
So Jack needs a total of 132 inches of rope.
Since 132 inches is less than the total length of rope that Jack has (252 inches).
Therefore, he does have enough rope for both projects.
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Emails arrive randomly on Oliver's computer at an average rate of 1.46 per hour. Stating a necessary assumption, find the probability that between 9.00am and 11.30am Oliver receives: (i) no emails
Answers
The probability that Oliver will receive no emails in between 9.00am and 11.30am is 2.599%.
What is Poisson Distribution?Poisson Distribution is defined as a discrete distribution which measures the likelihood of happening an event in a certain period of time.
The formula for Poisson distribution is,
P(x) = (e^(-λ) λˣ) / x!
Given that,
Average rate of arrival of emails in an hour = 1.46 per hour
Between 9.00 am and 11.30 am, there are 2.5 hours.
Average rate of arrival of emails in 2.5 hours = 2.5 × 1.46
= 3.65
So, λ = 3.65
x = 0 (since no emails.)
P(0) = (e^(-3.65) (3.65)⁰) / 0!
= e^(-3.65)
= 0.02599
= 2.599%
Hence the probability is 2.599%.
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"The graph of y=−3x+62
Answers
Answer:
-1/3x + 62/3
Step-by-step explanation:
Interchange the variables: y = -3x + 62
Swap the sides: -3y + 62
Move constant: -3y + 62 = x
Divide and simplify: -3y = x - 62
Answer: -1/3x + 62/3
Please help me with my answer. I'm stuck and really need help on it.
Answer:
-3 = slope, 62 = y intercept
Step-by-step explanation:
Mary goes to the store to buy Pokémon cards for her sister Cloey’s birthday. She has $4,032 and Pokémon card packs cost $4 each. How many Pokémon cards can she buy?
Answers
Answer:
To determine how many Pokémon cards Mary can buy, we need to divide the amount of money she has by the cost per card:
$4,032 ÷ $4 = 1,008
Mary can buy 1,008 Pokémon cards.
Mr. Gaspar wants to store a 12-foot-long pipe
in a tool closet. The closet has the shape of a
right rectangular prism with the dimensions
shown. Will the pipe fit? Show your work.
Answers
The longest pipe can fit in the prism is 11.7 ft, so, 12 ft long pipe does not fit.
What is the Pythagoras theorem?The Pythagoras theorem which is also referred to as the Pythagorean theorem explains the relationship between the three sides of a right-angled triangle. According to the Pythagoras theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides of a triangle.
From the given figure,
By considering base of the prism, we get
c²=6²+6²
c²=72
c=6√2 ft
Now, d²=c²+8²
d²=72+64
d²=136
d=√136
d= 11.7 ft
Hence, the longest pipe can fit in the prism is 11.7 ft, so, 12 ft long pipe does not fit.
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The length of a radius of a circle, measured in feet, is represented by the expression z + 3.6. The diameter of the circle is 11 4/5 ft.
What is the value of z?
Enter your answer as a decimal or mixed number in simplest form in the box.
Answers
Answer:
Step-by-step explanation:
The diameter of the circle is twice the length of the radius, so we can write:
2r = d
where r is the radius and d is the diameter.
Substituting the given diameter of 11 4/5 ft, we get:
2r = 11 4/5 ft
To find the length of the radius, we can divide both sides by 2:
r = (11 4/5) / 2
r = 5 9/10 ft
We also know that the length of the radius is z + 3.6, so we can set up the equation:
z + 3.6 = 5 9/10
Subtracting 3.6 from both sides, we get:
z = 5 9/10 - 3.6
z = 2 3/10
Therefore, the value of z is 2 3/10 or 2.3 as a decimal.
Answer: The correct answer is 2.3
Step-by-step explanation: Answer confirmed correct on test so no worries
Convert 14ft3 to fluid ounces. Round to the nearest hundredth if necessary. First fill in the blanks on the left hand side with two of the ratios given. Then write the answer.
Answers
14 ft³ is equal to 104,720 fluid ounces.
What is Unit of Measurement?A unit of measurement is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity.
We have to convert 14 ft³ to fluid ounces
we need to use the conversion factor: 1 ft³ = 7,480 fluid ounces
Multiplying both sides by 14, we get:
14 ft³ = 14 x 7,480 fluid ounces
= 104,720 fluid ounces
Therefore, 14 ft³ is equal to 104,720 fluid ounces.
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Kabir has three fuel containers. One has a capacity of 7 litres,
one has a capacity of 4 litres and one has a capacity of 3 litres.
The 7 litre container is full, the other two containers are empty.
None of the containers has a measurement scale.
How can Kabir transfer the fuel so that two of the containers contain
a volume of 2 litres each, and the other contains a volume of 3 litres,
vithout having to estimate?
Answers
The solution is, the capacity of the largest container that can fill the tanks an exact number of times is, 44 litres.
What is CGF?In mathematics, the greatest common divisor of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers x, y, the greatest common divisor of x and y is denoted {\displaystyle \gcd}.
here, we have,
We subtract the remainders from the amounts.
447 - 7 = 440
577 - 5 = 572
669 - 9 = 660
Now we need the GCF of 440, 572, 660
440 = 2³ × 5 × 11
572 = 2² × 11 × 13
660 = 2² × 3 × 5 × 11
GCF = 2² × 11 = 44
The greatest common factor is 44.
Answer: 44
Check:
447/44 = 10 reminder 7
577/44 = 13 remainder 5
669/44 = 15 remainder 9
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Please help me with this math question!
Answers
Step-by-step explanation:
To write the expression (-4-7i)+(-1-4i) in a+bi form, we need to simplify the expression by adding the real parts and the imaginary parts separately.
(-4-7i)+(-1-4i) = -4 - 1 - 7i - 4i = -5 - 11i
Therefore, (-4-7i)+(-1-4i) can be written in a+bi form as -5 - 11i.
Answer:
a= -24
b= 23
Step-by-step explanation:
(-4-7i)(-1-4i)
4+16i+7i-28
-24i+23i
Bob's Gift Shop sold 550 cards for Mother's Day. One salesman, Arun, sold 2% of the cards sold for Mother's Day. How many cards did Arun sell?
Answers
Answer:
55 cards
Step-by-step explanation:
PLS I BEG YALL TO GUVE ME THE CORRECT ANSWER
Answers
The x intercept is (2, 0) and the y intercept is (0, 8)
What is an intercept?x-intercept and y-intercept are the two main intercepts. The line's x-intercept and y-intercept are located where the line crosses the x and y axes, respectively.
As per the given data:
The function is [tex]f(x) = -3^x + 9[/tex].
For the x intercept, f(x) = 0:
[tex]0 = -3^x + 9\\3^x = 9\\x = 2[/tex]
The x intercept is 2.
For the y intercept, x = 0:
[tex]y = -3^0 + 9\\y = -1 + 9\\y = 8[/tex]
The y intercept is 8.
Hence, The x intercept is (2, 0) and the y intercept is (0, 8)
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An architect designs a rectangular flower garden such that the width is exactly two-thirds of the length. If 320 ft of antique picket fencing are to be used to enclose the garden for the length and width of the garden.
Answers
Answer:
The length of the garden would be 480 ft and the width would be 320 ft.
Step-by-step explanation:
Suppose a company has fixed costs of $49,400 and variable cost per unit of
4
9
x + 222 dollars,
where x is the total number of units produced. Suppose further that the selling price of its product is
2148 −
5
9
x dollars per unit.
(a) Find the break-even points. (Enter your answers as a comma-separated list.)
x =
(b) Find the maximum revenue. (Round your answer to the nearest cent.)
$
(c) Form the profit function P(x) from the cost and revenue functions.
Find maximum profit.
$
(d) What price will maximize the profit? (Round your answer to the nearest cent.)
$
Answers
(a) The break-even points are approximately 1,581 and 2,809 units produced.
(b) The maximum revenue is approximately $2,088,931.47.
(c) P(x) = -5/9x² + 919x - 49,400
(d) The price that will maximize the profit is approximately $1,829.17 per unit.
What is the algebraic equations?An algebraic expression is when we use numbers and words in solving a particular mathematical question.
(a) To find the break-even points, we need to find the values of x at which the total revenue equals the total cost. The total revenue is the product of the selling price and the number of units sold, while the total cost is the sum of the fixed cost and the variable cost per unit multiplied by the number of units sold. Thus, we have:
Total revenue = Selling price × Number of units sold = (2148 - 5/9x)x
Total cost = Fixed cost + Variable cost per unit × Number of units sold = 49,400 + (49x/9 + 222)x
Setting the total revenue equal to the total cost, we get:
2148x - 5/9x² = 49,400 + (49x/9 + 222)x
Multiplying both sides by 9, we get:
19332x - 5x² = 445,110
Simplifying, we get:
5x²- 19332x + 445,110 = 0
Using the quadratic formula, we get:
x = (19332 ± sqrt(19332² - 4(5)(445,110))) / (2(5))
x ≈ 2808.6 or x ≈ 1581.4
Hence, the break-even points are approximately 1,581 and 2,809 units produced.
(b) The revenue function is given by:
R(x) = (2148 - 5/9x)x
To find the maximum revenue, we can take the derivative of the revenue function with respect to x, set it equal to zero, and solve for x:
R'(x) = 2148 - 10/9x = 0
x = 19332/10 ≈ 1933.2
Substituting this value of x into the revenue function, we get:
R(1933.2) ≈ $2,088,931.47
Hence, the maximum revenue is approximately $2,088,931.47.
(c) The profit function P(x) is given by:
P(x) = R(x) - C(x) = (2148 - 5/9x)x - (49,400 + (49x/9 + 222)x)
Simplifying, we get:
P(x) = -5/9x² + 919x - 49,400
(d) To find the maximum profit, we can take the derivative of the profit function with respect to x, set it equal to zero, and solve for x:
P'(x) = -10/9x + 919 = 0
x = 8265
Substituting this value of x into the profit function, we get:
P(8265) ≈ $1,077,927.78
Therefore, the maximum profit is approximately $1,077,927.78.
To find the price that will maximize the profit, we need to substitute the value of x into the selling price function:
Selling price = 2148 - 5/9x = 2148 - 5/9(8265) ≈ $1,829.17
Hence, the price that will maximize the profit is approximately $1,829.17 per unit.
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Find the sum of the following arithmetic series:
Answers
Answer:
19107
Step-by-step explanation:
First, we need to find the common difference (d), which we can find by subtracting two consecutive terms. Thus, we can do d = 1 - (-3) = 4.
Now, we need to know how many terms are in the arithmetic series. We can find the number using nth term formula, which is
[tex]a_{n}=a_{1}+(n-1)*d[/tex], where a1 is the first term, n is the term position (e.g., 1st, 2nd).
Although we don't know the term position of 389, we can still plug it into the formula and solving for n will reveal it's term number and the total number of sums in the sequence:
[tex]389=-3+(n-1)*4\\392=(n-1)*4\\392=4n-4\\396=4n\\99=n[/tex]
Now, we can find the sum using the sum formula, which is
[tex]S_{n}=n/2*(a_{1}+a_{n})[/tex]
Sn can become S99 since there are 99 terms and we can let an = a99, which we know is 389.
Now, we must solve for S99:
[tex]S_{99}=\frac{99}{2}*(-3+389)\\ S_{99}=\frac{99}{2}*(386)\\ S_{99}=19107[/tex]
Jake gets to a party at to'dock having alredy baned off 25 calories throughout his normal day. After dancing at the party. he checked! and burned off 85 total for the day by 9o'clock.
Answers
The calories burned per hour is 30 calories per hour
How to determine the calories burned per hourFrom the question, we have the following parameters that can be used in our computation:
Calories at 7 = 25 calories
Calories at 9 = 85 calories
Using the above as a guide, we have the following:
Rate = Difference in calories/Difference in time
Substitute the known values in the above equation, so, we have the following representation
Rate = (85 - 25)/(9 - 7)
Evaluate
Rate = 30
Hence, the rate is 30 calories per hour
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Your teacher has requested that two rows of strips be placed around the entire room what is the total length of strips that is required
Answers
The Length of wooden strip based on the information is 106 cm.
How to calculate the lengthThe perimeter of a shape is simply the total length of the boundary that the shape has. It should be noted that in this case, it's gotten by adding all the length that the shape has.
Length of wooden strip = Perimeter of a rectangular table.
= 2(Length+Breadth)
= 2(32+21)
= 2(53)
= 106 cm
Length of wooden strip is 106 cm.
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What is the length of the wooden strip required to frame a photograph of length and breadth 32 cm and 21 cm respectively?
Can u do number 15 plsssssss
Answers
Answer:
y=x+5 x=3
Plug 3 into x
y=3+5
5+3=8
Step-by-step explanation:
find the measure of gpy
Answers
The value of ∠ GPY, given the angle measures on the figure, can be found to be 118 °
How to find the angle ?The angles ∠ GPY and ∠ KPR are congruent, which means that they are the same.
The angle ∠ GPY can therefore be found by equating both angles to themselves:
8 p + 94 = 5 p + 103
Collect like terms ;
8 p - 5 p = 103 - 94
3 p = 9
p = 9 / 3
p = 3
The value of ∠ GPY is therefore :
= 5 ( 3 ) + 103
= 15 + 103
= 118 °
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Can someone please help me with this math problem, ASAP?
Answers
The equation rewritten as a function of x is
[tex]y \ln \left(-2 x+x^2+2 y \right)=\ln (x)[/tex]
What is a mathematical function?Generally, In the field of mathematics, an assignment of one element from set Y to each and every element in set X is the definition of a function. This particular set, X, is referred to as the function's domain, and this particular set, Y, is referred to as the function's codomain.
Historically, functions have been seen as an idealization of the way in which one variable relies on another variable. Wikipedia
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Find dz/dt using chain rule
Answers
Using the chain rule, the value for dz/dt is obtained as [tex]dz/dt = 2(2t + (4 - t^2)e^{(4-t^2)}) ((1 - t)e^{(4-t^2)})[/tex].
What is chain rule?
The chain rule is the formula used to determine the derivative of a composite function, such as cos 2x, log 2x, etc. Another name for it is the composite function rule. Only composite functions are subject to the chain rule.
To find dz/dt using the chain rule, we need to differentiate each term of z with respect to t, then multiply by the derivative of the inner function with respect to t.
First, we have -
x = 2t
Taking the derivative with respect to t, we get -
dx/dt = 2
Next, we have -
y = 4 - t²
Taking the derivative with respect to t, we get -
dy/dt = -2t
Using the chain rule, we have -
[tex]dz/dt = (x + y)e^y \times (\frac{dx}{dt} + \frac{dy}{dt}e^y)[/tex]
Substituting x and y into this equation, we get -
[tex]dz/dt = (2t + (4-t^2))e^{(4-t^2)} \times ((2 - 2t) e^{(4-t^2)})[/tex]
Simplifying this expression, we get -
[tex]dz/dt = 2(2t + (4 - t^2)e^{(4-t^2)}) ((1 - t)e^{(4-t^2)})[/tex]
Therefore, the value is obtained as [tex]dz/dt = 2(2t + (4 - t^2)e^{(4-t^2)}) ((1 - t)e^{(4-t^2)})[/tex].
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