The area of the circle with diameter 6.4 is 32.15 square centimeters.
What is area of the circle?A circle closed plane geometric shape. In technical terms, a circle is a locus of a point moving around a fixed point at a fixed distance away from the point.
Area of a circle is the region occupied by the circle in a two-dimensional plane. It can be determined easily using a formula:
[tex]A = \pi r^2[/tex]
Given that the diameter of the circle is 6.4 centimeters.
The radius of the circle is thus half of the diameter that is 3.2 centimeters.
The area of the circle is given by the formula:
[tex]A = \pi r^2[/tex]
Substitute the value of r = 3.2 in the formula:
[tex]A = (3.14) (3.2)^2\\\\A = 32.15[/tex]
The area of the circle is 32.15 square centimeters.
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Give me 2 rational exponent world problem that contains sustainability
1. A city wants to reduce its carbon footprint by switching to renewable energy sources. The city's current energy consumption is measured in gigawatt hours (GWh) and they want to reduce it by 20% in the next 5 years. The city's current energy consumption is 2,000 GWh per year. What is the target energy consumption in GWh per year that the city should aim for to achieve their goal?
2. A company wants to reduce its water consumption by 50% in the next 10 years. The company currently consumes 500,000 gallons of water per day. How many gallons of water per day should the company aim to consume in 10 years to achieve their goal?
Use the graph of y=f(x) to
A. Determine f(-1)
B. Find the range of f.
(A) f(-1) = 1
(B) The range of the function f(x) is [-4, ∞).
What is the range of the function?
In mathematics, a function's range can refer to one of two closely related ideas: Its codomain is a function. The picture of the action Given two sets X and Y, a binary relation f between them is a function if and only if there is precisely one y in Y such that f connects x to y for every x in X.
For the given function we have to find the value of f(-1) and to find the range of the function.
From graph we can see that,
f(-1) = 1
The range of the function is the interval in which function cover the x - axis.
The range of the function is [-4, ∞).
Hence,
(A) f(-1) = 1
(B) The range of the function f(x) is [-4, ∞).
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There are 80 Students in the marching band.
Can they match in 2,5, and /or 10 equal rows? Explain.
Answer:
Step-by-step explanation: Yes, they can match in 2, 5, and/or 10 equal rows. This is because the number of items in the set can be divided evenly by 2, 5, and 10. For example, if there are 20 items in the set, it can be divided into 2 rows of 10, 5 rows of 4, or 10 rows of 2.
Which of the following are true statements about waves?
A. All waves carry matter but not energy.
B. All waves have both an amplitude and frequency.
C. All waves have a repeating pattern.
Answer:
B,C
Step-by-step explanation:
B. All waves have both an amplitude and frequency.
C. All waves have a repeating pattern.
A statement A is incorrect, Waves are a disturbance that transfer energy through space without transferring matter.
B is true, All types of waves have an amplitude, which is the measure of the maximum displacement of a point on the wave from its rest position and a frequency, which is the number of complete oscillations of a point on the wave per unit of time.
C is also true, All types of waves have a repeating pattern, which is known as a waveform. The shape of the waveform determines the type of wave. For example, sine waves have a smooth, curved shape, while square waves have sharp, angular edges.
You have a cone whose height and base radius are equal and, inside it, you put the largest possible sphere that will completely fit inside. What is the exact fraction of the volume of the cone that is occupied by the sphere?
The exact ratio of of the volume of the cone that is occupied by the sphere is 4/(1+√2)³.
What is a cone?
A cone is a three-dimensional geometric shape that narrows smoothly from a flat base (typically circular base) to a point called the apex or vertex (which creates an axis to the centre of base).
Given that the height and radius of the cone is the same.
Assume that the radius of the cone is x.
Thus the slant height of the cone is √(x² + x²) = x√2.
Assume the radius of the sphere is r.
△OEC and △BDC are similar triangles according to AAA rule.
Thus, BD/OE =BC/OC
x/r = x√2/(x -r)
Cross multiply:
x(x - r) = xr√2
x² - rx = xr√2
x² = xr√2 + rx
rx( 1+√2) = x²
r = x²/[x( 1+√2)]
r = x/ (1+√2)
The volume of the sphere is 4/3 ∏ r³ = 4/3 ∏ [ x/ (1+√2)]³
The volume of the cone is 1/3 ∏r²h = 1/3 ∏x³
The ratio of the volume of the sphere to the cone is
4/3 ∏ [ x/ (1+√2)]³ : 1/3 ∏r³
= 4 : (1+√2)³
= 4/(1+√2)³
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Merge onto Highway 40 and drive 3/5
mile. Stop and pay the toll. Then
continue on Highway 40 for twice this.
distance. How much longer will you be
on Highway 40 after you pay the foll?
Distance traveled after toll payment is 1.2 miles on highway 40.
What is Distance ?The distance may be calculated using a curved route. Displacement measurements can only be made along straight lines. Distance is path-dependent, meaning it varies depending on the direction followed. Displacement simply depends on the body's beginning and ending positions; it is independent of the route.
Distance is the sum of an object's movements, regardless of direction. Distance may be defined as the amount of space an item has covered, regardless of its beginning or finishing position.
The size or extent of the displacement between two points is referred to as distance. Keep in mind that the distance between two points and the distance traveled between them are not the same. The entire length of the journey taken between two points is known as the distance traveled. Travel distance is not a vector.
Distance traveled before toll payment =3/5 miles on highway 40
Distance traveled after toll payment =2*3/5 = 6/5 =
1.2 miles on highway 40.
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What is the graph of the solution to the following compound inequality?
7x+3 ≤ 52 and 3-x<9
O A.
O B.
О с.
O D.
-10-9-8-7 -6 -5 4 -3 -2 -1 0 1 2 3 4
5 6 7 8 9 10
-10-9-8-7 -6 -5 4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
-10-9-8-7 -6 -5 4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
B
-10-9-8-7 -6 -5 4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Here, on solving the provide question to us, we got to know that - equation 210 feet represents = 825/4
What is equation?Since equations are essentially questions, efforts to systematically find answers to those questions have been the driving force behind the development of mathematics. From straightforward algebraic equations that just require addition or multiplication to differential equations, exponential equations that use exponential expressions, and integral equations, there are many different types of equations that range in complexity.
Here,
15 feet represents = 1/8 of an inch
1 feet represents = 1/8 X 15
The growth of mathematics has been fueled by efforts to methodically find answers to equations, which are basically questions. There are many distinct forms of equations that range in complexity, from simple algebraic equations that just need addition or multiplication to differential equations, exponential equations that employ exponential expressions, and integral equations 210 feet represents = 1/8 X 15 X 210 = 825/4
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A capsule is in the form of a hollow cylinder surmounted by a hemispherical bowl of the same diameter, 12 cm on both ends. The total height of the capsule is 16 cm. Find the surface area of the capsule. п cm?
The surface area of the capsule is calculated by adding the surface area of the cylinder and the surface area of the hemispherical bowl.
What is the hemispherical bowl?A hemispherical bowl is a type of bowl with a rounded bottom that is shaped like half of a sphere. It is often used in laboratories, kitchens, and dining rooms. The bowl’s curved surface prevents the contents from spilling, which makes it an ideal choice for mixing, swirling, and scooping. Additionally, the bowl’s shape makes it look aesthetically pleasing, so it can be used as a decorative piece as well.
The surface area of the cylinder = 2πr2 + 2πrh
Where r is the radius of the cylinder and h is the height of the cylinder.
The radius of the cylinder = 12 cm
The height of the cylinder = 16 cm
Therefore, the surface area of the cylinder = 2π(12)2 + 2π(12)(16) = 1536π cm2
The surface area of the hemispherical bowl = 2πr2
Where r is the radius of the hemispherical bowl.
The radius of the hemispherical bowl = 12 cm
Therefore, the surface area of the hemispherical bowl = 2π(12)2 = 144π cm2
The total surface area of the capsule = 1536π cm2 + 144π cm2 = 1680π cm2
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There are some counters in a bag.
The counters are red or white or blue or yellow.
Bob is going to take at random a counter from the bag.
The table shows each of the probabilities that the counter will be blue or will
be yellow.
Colour
Probability
red white blue yellow
0.45. 0.25
There are 18 blue counters in the bag.
The probability that the counter Bob takes will be red is twice the probability that the
counter will be white.
(a) Work out the number of red counters in the bag.
The number of red counters in the bag is given as follows:
8.
How to obtain a probability?A probability is obtained by the division of the number of desired outcomes by the number of total outcomes.
The probabilities for this problem are given as follows:
Red: 2x.White: x.Blue: 0.45.Yellow: 0.25.The sum of the probabilities is of one, hence the value of x is obtained as follows:
2x + x + 0.45 + 0.25 = 1
3x = 0.3
x = 0.1.
Hence the proportion of red is given as follows:
0.2.
There are 18 blue counters in the bag, which is of 0.45 of the total amount, hence the total amount is obtained as follows:
0.45n = 18
n = 18/0.45
n = 40.
Meaning that the number of red counters is given as follows:
0.2 x 40 = 8.
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A toy car and a battery together costs £44. The ratio of the cost of the car to the battery is 10:1. How much did the battery cost?
Answer:
4,4
Step-by-step explanation:
44x1/10
8,320 to 10,399 114,192 10,400 to 15,599 148,276 15,600 to 20,799 123,638 20,800 to 25,999 121,623 26,000 to 31,199 103,402 31,200 to 36,399 73,463 36,400 to 41,599 59,126 41,600 to 51,999 68,747 52,000 to 77,999 56,710 1. What is the modal-class interval? 2. Copy the table into your notebook and include columns to find the upper pulative frequencies and cumulative percentages.
Answer:
Copy the table into your notebook and include columns to find the endpoint (upper value) for each interval. Figure out the cumulative frequency and
Step-by-step explanation:
Create Write a real-world problem that could be represented by the equation 12(x+2.50)=78 .
Algebraic expression explaining real world problem is - A store is having a sale where all items are marked down 12% and an additional $2.50 is taken off the final price.
If a customer wants to find out the original price of an item before the sale, they can use the equation 12(x+2.50)=78, where x is the original price of the item.
What are algebraic expressions?Algebraic expressions are known as expressions made up of :
VariablesConstantsAddition,SubtractionDivisionOther algebraic operationsWhat are some examples of algebraic expression?A mathematical statement with variables, constants, coefficients, and algebraic operations is known as an algebraic expression. A good example of an algebraic expression is 5x2+6xyc. Algebraic expressions do not use equal signs, in contrast to algebraic equations. In contrast, since they are not derived from integer constants and algebraic operations, transcendental numbers like and e are not algebraic.
A real-world problem:A store is having a sale where all items are marked down 12% and an additional $2.50 is taken off the final price. If a customer wants to find out the original price of an item before the sale, they can use the equation 12(x+2.50)=78, where x is the original price of the item.
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find the derivative of the function. f(x) = (5x6 + 4x3)4
The derivate of function f(x)=(5x⁶+4x³)⁴is F'(x) = 4(30x³+12x²)(5x⁶+4x³)³
The derivative of a function f(x) represents its rate of change and is denoted by f'(x) or df/dx.
The derivative Formula is given as:
f1 (x)=lim△x→0 f(x+△x) − f(x)△x.
Take the derivative: F'(x) = [tex]\frac{d}{dx} (5x^6+4x^3)^5[/tex]
Substitute the inner function by (5x⁶+4x³)
Apply the chain rule: [tex]\frac{d}{dx} (g)^4*\frac{d}{dx} (5x^6+4x^3)^4[/tex]
Calculate the derivative: [tex]4g^3*\frac{d}{dx} (5x^6+4x^3)^4[/tex]
Substitute back: [tex]4(5x^6+4x^3)^3*\frac{d}{dx} (5x^6+4x^3)[/tex]
Result after applying the chain rule: F'(x) = [tex]4(5x^6+4x^3)^3*\frac{d}{dx} (5x^6+4x^3)[/tex]
Calculate the derivative of the sum or difference:
F'(x) =[tex]4(5x^6+4x^3)^3*(\frac{d}{dx} (5x^6))+\frac{d}{dx} (4x^3))[/tex]
Take out the coefficients F'(x) = [tex]4(5x^6+4x^3)^3*(5*\frac{d}{dx} (x^6)+4\frac{d}{dx}(x^3))[/tex]
Apply the power rule: F'(x) = 4(5x⁶+4x³)³*(5*6x³+4*3x²)
Multiply the monomials: F'(x) = 4(5x⁶+4x³)³*(30x³+12x²)
Rewrite the expression: F'(x) = 4(30x³+12x²)*(5x⁶+4x³)³
So, the answer is F'(x) =4(30x³+12x²)*(5x⁶+4x³)³
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4. A model kit for the Dornier Do17 airplane has a wingspan of approximately 10 inches. If the kit has a
scale of 1 inch to 72 inches, what is the wingspan of the actual Dornier Do17? Show all your work.
Answer: If the model kit for the Dornier Do17 airplane has a wingspan of 10 inches and the scale of the kit is 1 inch to 72 inches. This means that for every inch on the model, the equivalent measurement on the actual Dornier Do17 airplane is 72 inches.
We know that the wingspan of the model is 10 inches and we want to find the actual wingspan of the airplane so we'll use the scale factor of 1:72
To find the actual wingspan, we'll multiply the wingspan of the model by the scale factor.
10 inches * 72 inches/1 inch = 720 inches
Therefore, the actual wingspan of the Dornier Do17 airplane is 720 inches or 60 feet.
Step-by-step explanation:
The two-way frequency table contains data on the preference of two school subjects among a group of 100 students. English Science Total Grade 5 24 36 60 Grade 8 16 24 40 Total 40 60 100 Based on this data, are "Grade 5" and "Science" independent events? (1 point) No, P(Grade 5 ∩ Science) ≠ P(Grade 5) ⋅ P(Science) Yes, P(Grade 5 ∩ Science) = P(Grade 5) ⋅ P(Science) No, P(Grade 5 ∩ Science) = P(Grade 5) ⋅ P(Science) Yes, P(Grade 5 ∩ Science) ≠ P(Grade 5) ⋅ P(Science)
The events Grade 5 and Science" are independent events, P(Grade 5 ∩ Science) = P(Grade 5) ⋅ P(Science) .Option B is the correct option.
What is frequency?
The frequency (f) of a specific value is the number of occurrences of the value in the data. The frequency distribution of a variable is the collection of all possible values and the frequencies associated with these values. Frequency distributions are represented graphically as frequency tables or charts.
Given table is
English Science Total
Grade 5 24 36 60
Grade 8 16 24 40
Total 40 60 100
Total students is 100.
Total student in grade 5 is 60.
The number of students who prefer science is 60.
The number of students who are in grade 5 and prefer science is 36.
The probability of grade 5 is P(grade 5) = 60/100 =0.6
The probability of science is P(science) = 60/100 = 0.6
The probability of Grade 5 ∩ Science is P(Grade 5 ∩ Science) = 36/100 =0.36.
Thus P(Grade 5 ∩ Science) =P(Grade 5) ⋅ P(Science).
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NO LINKS!!!
a. Identify each sequence as arithmetic, geometric, or neither.
b. If it is arithmetic or geometric, describe the sequence generator
n t(n)
0 16
1 8
2 4
3 2
4 1
5 1/2
6 1/4
Answer:
a) The sequence is geometric,b) [tex]t(n) = 2^{4-n}[/tex]----------------------------------------
Given sequence from zeros to 6th terms:
16, 8, 4, 2, 1, 1/2, 1/4It is a GP with the common ratio of 1/2 as each term is half the previous one.
Use the nth term equation, considering the first term is t(1) = 8, and common ratio r = 1/2:
[tex]t(n) = t(1)*r^{n-1}[/tex][tex]t(n) = 8*(1/2)^{n-1}=8*2^{1-n}=2^3*2^{1-n}=2^{4-n}[/tex]Answer:
a) Geometric sequence
[tex]\textsf{b)} \quad t(n)=8 \left(\dfrac{1}{2}\right)^{n-1}[/tex]
Step-by-step explanation:
An Arithmetic Sequence has a constant difference between each consecutive term.
A Geometric Sequence has a constant ratio (multiplier) between each consecutive term.
Part (a)From inspection of the given table, t(n) halves each time n increases by 1.
Therefore, the sequence is geometric with a common ratio of 1/2.
Part (b)[tex]\boxed{\begin{minipage}{5.5 cm}\underline{Geometric sequence}\\\\$a_n=ar^{n-1}$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the first term. \\\phantom{ww}$\bullet$ $r$ is the common ratio.\\\phantom{ww}$\bullet$ $a_n$ is the $n$th term.\\\phantom{ww}$\bullet$ $n$ is the position of the term.\\\end{minipage}}[/tex]
Given:
a = 8r = 1/2aₙ = t(n)Substitute the initial value (when n = 1) and the common ratio into the formula to create an equation for the nth term:
[tex]\implies t(n)=8 \left(\dfrac{1}{2}\right)^{n-1}[/tex]
what is the average (mean) value of 3t^3-t^2 over the interval -1<=t<=2?
The average value of 3t^3-t^2 over the interval -1<=t<=2 is 11/4
Mean Value Theorem states if a function f(x) is continuous on the interval [a,b] and differentiable on (a,b), there is at least one value c in the interval (a<c<b) such that:
f '(c)=f(b)-f(a)/b-a
Average Value:
The average value of a function of a variable over a range represents the height of a rectangle of the same area as that defined by the function under the curve. This value can be calculated using a formula:
favg=1/b−a∫f(t)dt
consider f(x)=3t³-t² and limits are -1 to 2
Now, substitute these values in the above formula:
favg=1/2-(1)∫3t³-t²dt
favg=1/3∫(3/4t⁴-1/3t³)|₋₁²
favg=(12-8/3)-(3/4+1/3)
By solving the equation we get
favg=11/4
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Find four pairs of integers that add up to 1.
Answer:
5 & -4
4 & -3
3 & -2
2 & -1
Step-by-step explanation:
5 + (-4) = 1
4 + (-3) = 1
3 + (-2) = 1
2 + (-1) = 1
if x/y=z/w then show that x^m+y^m+z^m+w^m/x^m+y^-m+z^-m+w^-m= (xyzw)^m/2
On solving the provided question we can say that the inequality equation can solved as = [tex]\sqrt[4]{xyzw}[/tex]
What is inequality?An inequality in mathematics is a relationship between two expressions or values that is not equal. Thus, imbalance leads to inequality. An inequality creates the link between two values that are not equal in mathematics. Egality is distinct from inequality. When two values are not equal, most commonly use the not equal sign (). Different inequalities are used to contrast values, no matter how little or large. Many simple inequalities may be resolved by modifying the two sides until the variables are all that remain. But a number of things contribute to inequality: Negative values on both sides are divided or added. Trade off the left and right.
here,
the inequality equation can solved as
[tex]x^m+y^m+z^m+w^m/x^m+y^-m+z^-m+w^-m= (xyzw)^{m/2}[/tex]
[tex]u = x+y/2 \geq \sqrt{xy}[/tex]
v = z+w /2 [tex]\geq \sqrt{zw}[/tex]
u+v/2 = x+y+z+w+/4 [tex]\geq \sqrt{uw} \geq \sqrt{\sqrt{xy} \sqrt{zw} }[/tex]
= [tex]\sqrt[4]{xyzw}[/tex]
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The sales tax in the town where Gina lives is 8%. Gina wants to buy a table saw that costs $300. How much sales tax will she pay?
Find a set of four integers such that their order and the order of their absolute valúes are the same
The set of 4 integers are is given by the equation A = { 1 , 2 , 3 , 4 }
What is an Equation?Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given data ,
Let the set of 4 integers be represented as set A
Now , the equation will be
Let the first number be 1
The absolute value of 1 is | 1 | = 1
Let the second number be 2
The absolute value of 2 is | 2 | = 2
Let the third number be 3
The absolute value of 3 is | 3 | = 3
Let the fourth number be 4
The absolute value of 4 is | 4 | = 4
And , | 1 | < | 2 | < | 3 | < | 4 |
Hence , the set of integers are { 1 , 2 , 3 , 4 }
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What is the slope for this
Answer:
-3
Step-by-step explanation:
rise over run method
3 up and 1 to the left which makes the slope negative
Find the range of the given function y=3x-2 for the domain (-1,2,4)
The range of the given function y = 3x - 2 for the domain (-1, 2, 4) is
What is Function?A function is a relation from a set A to a set B where the elements in set A only maps to one and only one image in set B. No elements in set A has more than one image in set B.
Domain of the function is the values of x for which the function is defined.
Range of the function is the value of the function when we put the values in the domain.
Given function,
y = 3x - 2
Given the domain (-1, 2, 4)
x = -1, then y = (3 × -1) - 2 = -5
x = 2, then y = (3 × 2) - 2 = 4
x = 4, then y = (3 × 4) - 2 = 10
Range is (-5, 4, 10).
Hence the range of the function is (-5, 4, 10).
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Which of the following describes the domain and range of the function?
The domain and range of the piecewise function is;
B) Domain: x < 2 or x ≥ 3.; range: y > -3
What is the domain and range of the function?The piecewise function is given as;
f(x) = {-¹/₂x - 2, if x < 2
{2x - 4, if x ≥ 3
The domain of a function is the set of all possible input values of the function and as such in this case the domain is x < 2 or x ≥ 3.
The range is defined as the set of all possible output values of the function and in this case, we have;
For -¹/₂x - 2; If x < 2
At x = 2; -¹/₂(2) - 2 = -3
Range is; y > -3
Similarly, for 2x - 4, if x ≥ 3
At x = 3; 2(3) - 4 = 2
Range is; y ≥ 2
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solve for x, given the rectangle below TNREO to= 3x+1 ro=2x+4
Answer:
Step-by-step explanation:
The length of the two diagonals of a rectangle are said to be equivalent.
From the Pythagorean theorem, the diagonal (a) will be the hypotenuse in the equation and the length (b) and width (c) will be its opposite and adjacent.
From the figure given, the length of half of two diagonal of rectangles are given.
We can simply solve for the value of x
Solution:TO = RO
3x + 1 = 2x + 4 (combine like terms by first transposing it to the other side of the equation and changing its sign from (+) to (-) or vice versa).
3x - 2x = 4 - 1
x = 3
Given the coordinates below, what is the scale factor of a dilation centered at the origin where "overline"(A'B') is the image of "overline"(A'B')
A(6,-4), B(2,-8)
A'(9,-6), B'(3, -12)
A.1/2
B.3/2
C.2/3
D. 3
Considering the coordinates of the preimage of line AB represented by A (6, -4), B (2, -8) and coordinates of the image of line A'B' represented by A' (9, -6), B'(3, -12) the dilation factor is 3/2
How to find the dilation factorThe transformation rule for dilation is as follows
(x, y) for a scale factor of r → (rx, ry)
following similar procedure for the given problem we have
A (6, -4) transformed to A' (9, -6)
B (2, -8) transformed to B' (3, -12)
hence 6r = 9, and -4r = -6
r = 9/6 = -6/-4
r = 3/2
The dilation factor of the transformation is 3/2
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4) Daniella and Brayan have a joint checking account. They have a balance of $3,839.25 in the check register. The balance on the bank statement is $3,450.10. Not reported on the statement are deposits of $2,000, $135.67, $254.77, and $188.76 and four checks for $567.89, $23.83, $598.33, and $1.000. Reconcile the bank statement.
End Balance: $3,839.25 + $2,579.20 - $2,190.05 = $5,735.15, so the reconciled balance of $5,735.15.
What is bank reconciliation theory?This theory is used to compare the bank statements to the check register in order to properly reconcile the two balances.
The theory used in this question is the bank reconciliation theory.
Step 1: Begin by looking at the two balances - the check register balance and the bank statement balance.
In this case, the check register balance is $3,839.25 and the bank statement balance is $3,450.10.
Step 2: Add up all of the deposits that are not reported on the bank statement. In this case, the deposits are $2,000, $135.67, $254.77, and $188.76.
Step 3: Subtract all of the checks that are not reported on the bank statement. In this case, the checks are $567.89, $23.83, $598.33, and $1,000.
Step 4: Add the deposits and subtract the checks from the check register balance.
In this case, the check register balance ($3,839.25) plus the deposits ($2,579.20) minus the checks ($2,190.05) gives a reconciled balance of $5,735.15.
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"Find all the second partial derivatives
f(x,y)= sin^(2) (mx+ny)"
Therefore , the solution to the given problem of derivative comes out to be 2cos(mx+ny) for both the cases of partial derivative.
Define derivative.The definition of a derivative is the simultaneous change that occurs at a certain place. Implicit & explicit functions are typically distinguished from one another. Explicit functions are those where the value of the known individual entity "x" directly determines the value of the known dependent variable "y."
Here,
Given : f(x,y)= sin^(2) (mx+ny)"
Partial derivative with respect to x
=> dx/dy = 2sin (mx+ ny) *m
=> d²x/dy = 2cos(mx+ny)
Partial derivative with respect to y
=> dx/dy = 2sin (mx+ ny) *n
=> d²x/dy = 2cos(mx+ny)
Therefore , the solution to the given problem of derivative comes out to be 2cos(mx+ny) for both the cases of partial derivative.
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Find two positive numbers whose product is 192 and whose sum is a minimum.
A. 3 and 64
B. 4[tex]\sqrt{3}[/tex] and 16[tex]\sqrt{3}[/tex]
C. 8 and 24
D. 8[tex]\sqrt{3}[/tex] and 8[tex]\sqrt{3}[/tex]
E. 12 and 16
Two positive numbers whose product is 192 and whose sum is a minimum are; 8√3 and 8√3.
The correct option is (D).
What is minima?Minima is the minimum value of a function in given domain.
First Order Derivative Test
Let f be the function defined in an open interval I and f be continuous at critical point c in I such that f’(c) = 0.
If f’(x) changes sign from negative to positive as x increases through point c, then c is the point of local minima. And the f(c) is the minimum value.
Given,
Product of two positive numbers = 192
Sum should be minimum
Let x be the first positive number and y be the other positive number. So, the equation would be
x . y = 192
⇒ y = 192/x --------(a)
Let S be the sum of the two positive numbers.
S = x + y
substituting value of y in terms of x from equation(a)
S = x + 192/x
⇒ S = x + 192 . x⁻¹
Differentiating with respect to x
S'(x) = 1 + (-1)192x⁻²
S'(x) = 1 - 192/x²
S'(x) = (x² - 192)/x²
To determine the minimum S, equating first derivative of S with respect to x to zero.
S'(x) = 0
(x²-192)/x² = 0
x² - 192 = 0
x² = 192
x = ±√192
Rejecting negative root, since the numbers are positive
x = √192
Substituting value of x in equation (a)
y = 192/√192
y = √192
x and y can be written as
x = √192 = 8√3
y = √192 = 8√3
Hence , 8√3 and 8√3 are the two positive numbers whose product is 192 and sum is a minimum.
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NO LINKS!!
Given the function: 1/(x + 1) - 4:
a. State the Domain using INTERVAL notation
b. State the Range using INTERVAL notation
c. Are there any asymptotes? If so, state their equations and draw them on the graph as dotted lines
Answer:
a) (-∞, -1) ∪ (-1, ∞)
b) (-∞, -4) ∪ (-4, ∞)
c) Vertical asymptote: x = -1
Horizontal asymptote: y = -4
Step-by-step explanation:
Given function:
[tex]f(x)=\dfrac{1}{x+1}-4[/tex]
Part (a)The domain of a function is the set of all possible input values (x-values).
When the denominator of a rational function is zero, the function is undefined.
Rewrite the function as one fraction:
[tex]\implies f(x)=\dfrac{1}{x+1}-\dfrac{4(x+1)}{x+1}[/tex]
[tex]\implies f(x)=\dfrac{1-4(x+1)}{x+1}[/tex]
[tex]\implies f(x)=\dfrac{-4x-3}{x+1}[/tex]
Set the denominator to zero and solve for x:
[tex]\implies x+1=0[/tex]
[tex]\implies x=-1[/tex]
Therefore, the given function is undefined when x = -1, so its domain in interval notation is:
(-∞, -1) ∪ (-1, ∞)Part (b)The range of a function is the set of all possible output values (y-values).
As the domain is restricted to (-∞, -1) ∪ (-1, ∞), the range is also restricted.
To find the range of a rational function, first solve the equation for x:
[tex]\implies y=\dfrac{1}{x+1}-4[/tex]
[tex]\implies y+4=\dfrac{1}{x+1}[/tex]
[tex]\implies (y+4)(x+1)=1[/tex]
[tex]\implies x+1=\dfrac{1}{y+4}[/tex]
[tex]\implies x=\dfrac{1}{y+4}-1[/tex]
[tex]\implies x=\dfrac{1-(y+4)}{y+4}[/tex]
Set the denominator of the resultant equation ≠ 0 and solve for y:
[tex]\implies y+4 \neq 0[/tex]
[tex]\implies y \neq -4[/tex]
Therefore, the range is the set of all real numbers other than y = -4:
(-∞, -4) ∪ (-4, ∞)Part (c)A vertical asymptote occurs at the x-value(s) that make the denominator of a rational function zero.
Therefore, there is a vertical asymptote at x = -1.
As the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the result of dividing the highest degree term of the numerator by the highest degree term of the denominator.
[tex]\implies f(x)=\dfrac{-4x-3}{x+1}[/tex]
Therefore, there is a horizontal asymptote at:
[tex]y=\dfrac{-4}{1}=-4[/tex]There are no slant asymptotes as there is a horizontal asymptote.