We write as an inverse proportion first then make an equation by multiplying by k:
[tex]y=\frac{k}{x}\Rightarrow k=x\times y[/tex]Find the value of k:
[tex]k=7\times12=84[/tex]Then, when x = 2, y is:
[tex]y=\frac{84}{2}=42[/tex]Answer: y = 42
Find the derivativef(x) = 1 / (x - 2)
ANSWER
[tex]\frac{df}{dx}=-\frac{1}{(x-2)^2}[/tex]EXPLANATION
We want to find the derivative of the given function:
[tex]f(x)=\frac{1}{x-2}[/tex]First, we have to rewrite the function as follows:
[tex]f(x)=(x-2)^{-1}[/tex]Next, make the following substitution:
[tex]a=x-2[/tex]The function now becomes:
[tex]f(x)=a^{-1}[/tex]Apply the chain rule of differentiation:
[tex]\frac{df}{dx}=\frac{df}{da}\cdot\frac{da}{dx}[/tex]Therefore, we have that:
[tex]\frac{df}{da}=-1\cdot a^{-1-1}=-a^{-2}[/tex]and:
[tex]\frac{da}{dx}=1[/tex]Therefore, the differentiation of the function is:
[tex]\begin{gathered} \frac{df}{dx}=-a^{-2}\cdot1 \\ \Rightarrow\frac{df}{dx}=-(x-2)^{-2}\cdot1 \\ \frac{df}{dx}=-\frac{1}{(x-2)^2} \end{gathered}[/tex]Which statement describes how the reasonable domain
compares to the mathematical domain?
A statement which best describes how the reasonable domain compares to the mathematical domain is that: C. the mathematical domain includes all real numbers, while the reasonable domain includes only real numbers greater than 2.
What is a domain?In Mathematics, a domain can be defined as the set of all real numbers for which a particular function is defined. This ultimately implies that, a domain is the set of all possible input numerical values or numbers to a function and the domain of any graph comprises all the input numerical values or numbers which are primarily shown on the x-axis.
Next, we would evaluate the function which represents the perimeter of this rectangle by substituting the value of 2 as follows:
f(w) = 6w – 8
f(2) = 6(2) – 8
f(2) = 12 – 8
f(2) = 4.
For the length of this rectangle, we have:
Length = 2w - 4
Length = 2(2) - 4
Length = 4 - 4
Length = 0
Therefore, the width of this rectangle must be real numbers that are greater than 2.
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Complete Question:
A rectangle has a length that is equal to 4 less than twice the width. The function for the perimeter depending on the width can be expressed with the function f(w) = 6w – 8, where w is the width of the rectangle in centimeters.
Which statement describes how the reasonable domain compares to the mathematical domain?
Both the mathematical and reasonable domains include only positive real numbers.
Both the mathematical and reasonable domains include only positive whole numbers.
The mathematical domain includes all real numbers, while the reasonable domain includes only real numbers greater than 2.
The mathematical domain includes all real numbers, while the reasonable domain includes only whole numbers greater than 2.
Justin earns a base salary of $1500 per month at
the jewelry shop. He also earns a 3% commission
on all sales. If Just sold $82,975 worth of jewelry
last month, how much would he make for the
month including his base salary and commission?
The total salary earned by the Justin including the commission is $3959.25.
What exactly is the commission?A commission is extra money earned based on job performance.Members believe to be paid a specified amount of money depending on achievement marketed, sessions closed, recruits placed, and so on—when they agree to a commission-based position as well as commission framework (more by signing the agreement).For the given question;
The base salary of Justin is $1500.
The commission earned by Justin is 3%.
The jewelry of worth $82,975 last month.
Let x be the total commission earned.
Then,
3% of 82,975 = x
x = 3× 82,975 / 100
x = $2489.25
Total salary = base salary + commission
Total salary = $1500 + $2489.25
Total salary = $3959.25
Thus, the total salary earned by the Justin including the commission is $3959.25.
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Finding the area of unusual shapes
The shape in question is a composite shape.
It comprises two(2) shapes which are a triangle and a semi-circle.
The area of the shape is the sum of the area of the triangle and that of the semi-circle
The area of the triangle is:
[tex]A_{triangle}=\frac{1}{2}\times base\times height[/tex][tex]\begin{gathered} \text{Base of the triangle =}6\text{ yard} \\ Height\text{ of the triangle= 4 yard} \end{gathered}[/tex]Thus,
[tex]\begin{gathered} A_{triangle}=\frac{1}{2}\times6\times4 \\ A_{triangle}=12\text{ yards} \end{gathered}[/tex]Area of the Semi-circle is:
[tex]A_{semi-circle}=\frac{\pi\times r^2}{2}[/tex][tex]\begin{gathered} \text{Diameter of the circle=6 yard} \\ \text{Radius}=\frac{Diameter}{2} \\ \text{Radius}=\frac{6}{2}=3\text{ yard} \end{gathered}[/tex][tex]\begin{gathered} A_{semi-circle}=\frac{3.14\times3^2}{2} \\ A_{semi-circle}=\frac{28.26}{2} \\ A_{semi-circle}=14.13\text{ yard} \end{gathered}[/tex]Hence, the area of the composite shape is:
[tex]\begin{gathered} \text{Area of the triangle + Area of the semi-circle} \\ 12+14.13=26.13\text{ yard} \end{gathered}[/tex]In the following diagram, we know that line AB is congruent to line BC and angle 1 is congruent to angle 2. Which of the three theorems (ASA, SAS, or SSS) would be used to justify that triangle ABC congruent triangle CDA?
The theorem that justifies why triangle ABC is congruent to triangle CDA is the: SAS.
What is the SAS Theorem?The SAS theorem states that if we can show that two triangles have a pair of corresponding congruent included angles, and two pairs of corresponding sides that are also congruent to each other, then we can prove that both triangles are congruent to each other.
The triangles, ABC and CDA have:
Two pair of corresponding sides that are congruent to each other, which are AB ≅ BC, and AC ≅ CA.
A pair of corresponding included angles, which is angle 1 ≅ angle 2.
Based on the above known information, we can conclude that triangle ABC is congruent to triangle CDA by SAS theorem.
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follow the order of operations and evaluate the exponential expression (5+1)^2-(9-5)^2x3
Find the common difference of the arithmetic sequence 5,14,23
The Solution.
The given sequence is
[tex]5,14,23[/tex]The common difference of the arithmetic sequence is given by the formula below:
[tex]\text{common difference(d)=T}_2-T_1=T_3-T_2[/tex]In this case,
[tex]T_1=5,T_2=14,T_3=23[/tex]Substituting these values in the formula above, we get
[tex]\begin{gathered} \text{common difference = 14-5=23-14}=9 \\ \text{common difference}=9 \end{gathered}[/tex]So,the correct answer is 9.
simplify the following expression using the distributive property and combining like terms:3(y-4) -5(y+8)
3(y-4) - 5(y+8)
3y - 12 - 5y - 40
-2y - 52
1 x(x-2) how I do this
x(x-2)
Apply the distributive property:
x(x)+x(-2)
x^2-2x
A Ferris wheel at a carnival has a diameter of 72 feet. Suppose a passenger is traveling at 5 miles per hour. (A useful fact: =1mi5280ft.)
(a) Find the angular speed of the wheel in radians per minute.
(b) Find the number of revolutions the wheel makes per hour. (Assume the wheel does not stop.)
a) The Ferris wheel has an angular speed is 12.222 radians per minute.
b) The Ferris wheel makes 116.712 revolutions in an hour.
How to understand and analyze the kinematics of a Ferris wheel
Kinematics is a branch of mechanical physics that studies the motion of objects without considering its causes. In other words, kinematics studies displacements, velocities and accelerations in translation, rotation and combined motion. In this case we find a Ferris wheel rotating around its axis at constant rate.
a) Then, the angular speed (ω), in radians per minute, is determined by the following product:
ω = v / R
Where:
v - Linear velocity at the rim of the Ferris wheel, in feet per second.R - Radius of the Ferris wheel, in feet.Please notice that the length of the radius is the half of the length of the diameter.
If we know that v = 5 mi / h and R = 36 feet, then the angular speed of the wheel is:
ω = [(5 mi / h) · (1 h / 60 min) · (5280 ft / 1 mi)] / [(0.5) · (72 ft)]
ω = 12.222 rad / min
The angular speed is 12.222 radians per minute.
b) A revolution is equal to an angular displacement of 2π radians and an hour is equal to 60 minutes. Then, we can derive the number of revolutions in an hour by dimensional analysis:
n = (12.222 rad / min) · (1 rev / 2π rad) · (60 min / h)
n = 116.712 rev / h
There are 116.712 revolutions in an hour.
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3. A square has sides of length 612 inches. Area of length times width.
What is the area of the square in square inches?
The area of the square in square inches is 374544 inches².
How to find the area of a square?A square is a quadrilateral with 4 sides equal to each other. Opposite sides of a square is parallel to each other.
Each angle of a square is 90 degrees.
Therefore,
area of square = l²
where
l = side lengthTherefore,
The square has a side length of 612 inches. The area of the square can be found as follows:
area of square = l²
l = 612 inches
Hence,
area of square = 612²
Therefore,
area of the square = 612 × 612
area of the square = 374544 inches²
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Identify the domain and range of the relation. Is the relation a function? Why or why not?
{(-3, 1), (0, 2), (1, 5), (2, 4), (2, 1)}
Domain: {-3, 0, 1, 2}
Range: {1, 2, 4, 5}
The relation is not a function because one of its x-values has two corresponding y-values.
What is the Domain and Range of a Relation?All the set of values of x in a relation are referred to as the range of a relation, while all the set of values of y in a relation are called the domain of the relation.
How to Determine if a Relation is a Function?If each of the x-values in a relation all have only one possible corresponding y-value, then the relation is a function.
Given the relation, {(-3, 1), (0, 2), (1, 5), (2, 4), (2, 1)}:
The domain is: {-3, 0, 1, 2}
The range is: {1, 2, 4, 5}
The relation has two y-values, 4 and 1, that corresponds to the x-value, 2. Therefore, it is not a function.
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The given relation is not a function because its x-values have two corresponding y-values. Domain: {-3, 0, 1, 2} and Range: {1, 2, 4, 5}
What is the Domain and Range of a Relation?The domain of a function is the set of all the possible input values that are valid for the given function.
The range of a function is the set of all the possible output values that are valid for the given function.
Given the relation as {(-3, 1), (0, 2), (1, 5), (2, 4), (2, 1)}
Therefore,
The domain will be: {-3, 0, 1, 2}
The range will be: {1, 2, 4, 5}
The relation has two y-values, 4 and 1, which corresponds to the x-value, 2.
The given relation is not a function because its x-values have two corresponding y-values. Domain: {-3, 0, 1, 2} and Range: {1, 2, 4, 5}
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Jordan’s of Boston sold Lee Company of New York computer equipment with a $7,000 list price. Sale terms were 4/10, n/30 FOB Boston. Jordan’s agreed to pay the $400 freight. Lee pays the invoice within the discount period. What does Lee pay Jordan’s?
There are 17 girls at a party with 30 guests. What fraction
of the party guests are girls?
Answer:
17/30
Step-by-step explanation:
There are 17 girls at a party with 30 guests. What fraction of the party guests are girls?
17/30
Hi, I always have a hard time with these, the whole question is in the picture.
we have that
x1 ----> number of cars with a 7,000 gal capacity
x2 ---> number of cats with a 14,000 gal capacity
x3 ---> number of cars with a 28,000 gal capacity
so
7000x1+14000x2+28000x3=462000
using a 3x3 system of equations solver
the solution is
x1=-2s-4t+66
x2=s
x3=t
The answer is option B
Writing a equation of a circle centers at the origin
ANSWER
[tex]x^2+y^2=100[/tex]EXPLANATION
The general equation of a circle is:
[tex](x-h)^2+(y-k)^2=r^2[/tex]where (h, k) = the center of the circle
r = radius of the circle (i.e. distance from any point on its circumference to the center of the circle)
The center of the circle is the origin, that is:
[tex](h,k)=(0,0)[/tex]To find the radius, apply the formula for distance between two points:
[tex]r=\sqrt[]{(x_1-h)^2+(y_1-k)^2_{}}[/tex]where (x1, y1) is the point the circle passes through
Hence, the radius is:
[tex]\begin{gathered} r=\sqrt[]{(0-0)^2+(-10-0)^2}=\sqrt[]{0+(-10)^2} \\ r=\sqrt[]{100} \\ r=10 \end{gathered}[/tex]Hence, the equation of the circle is:
[tex]\begin{gathered} (x-0)^2+(y-0)^2=(10)^2 \\ \Rightarrow x^2+y^2=100 \end{gathered}[/tex]Evaluate 1312e 4 Sov? 3x²x3 dx (Type an exact answer.)
We have to solve the integral:
[tex]\int ^4_03x^2e^{x3}dx[/tex]We will apply a variable substitution in order to simplify the solution. We have a hint when we see that the derivative of x^3 is 3x^2, that is part of the factors.
[tex]\begin{gathered} u=x^3\Rightarrow du=(3x^2)dx \\ x=0\Rightarrow u=0^3=0 \\ x=4\Rightarrow u=4^3=64 \end{gathered}[/tex]Then, we can write:
[tex]\int ^4_03x^2e^{x3}dx=\int ^4_0e^{x3}(3x^2)dx=\int ^{64}_0e^udu[/tex]Then, we have a simpler integral to solve:
[tex]\int ^{64}_0e^udu=e^u+C=e^{64}-e^0=e^{64}-1[/tex]The exact solution is e^64-1.
In may 2020, there were 2,119,800 people in the available labor force in oregon. the unemployment rate in oregon for may 2020 was 14.2%. determine the number of people who were unemployed in oregon during may 2020. round your answer to the nearest whole number.
301,012
1) Considering the data, we can write out the following equivalence:
[tex]\begin{gathered} 2,119,800---100\% \\ x--------14.2\% \end{gathered}[/tex]2) Notice that we need to rewrite those figures as 100% =1, and 14.2% as 0.142.
Now we can calculate how many people are equivalent to that rate of unemployment:
[tex]\begin{gathered} \frac{2119800}{x}=\frac{1}{0.142} \\ x=2119800\cdot0.142 \\ x=301,011.6\approx301,012 \end{gathered}[/tex]Notice that we rounded off to the nearest whole number.
3) So, there were approximately 301,012 Oregonians unemployed at that time
Write each of the following products (the result to a multiplication problem) using exponents to express the results in a simpler form.(3a)(5a) __________(5p)(2p) ___________(3 inches)(5 inches)___________(5 feet)(2 feet)_________
Let's do the mutiplications:
(3a)(5a) = 15a²
(5p)(2p) = 10p²
(3 inches)(5 inches) = 15 inches²
(5 feet)(2 feet) = 10 feet²mutiplic
You deposit $6000 in an account earning 6% interest compounded continuously. How much will you have in the account in 10 years?
Solution
Step 1:
Write the compounded interest continuously formula.
[tex]\text{A = Pe}^{rt}[/tex]Step 2:
Given data
P = $6000
r = 6% = 0.06
t = 10 years
Step 3:
Substitute in the formula
[tex]\begin{gathered} A\text{ = Pe}^{rt} \\ A\text{ = 6000 }\times\text{ 2.7183}^{10\times0.06} \\ A\text{ = 6000 }\times\text{ 2.7183}^{0.6} \\ A\text{ = 6000 }\times\text{ 1.822126} \\ A\text{ = \$10932.76} \end{gathered}[/tex]Final answer
A = $10933 ( nearest whole number)
Solve the following system of equations by graphing. Graph the system below and enter the solution set as an ordered pair in the form (x,y).if there are no solutions enter none and inter all if there are infinite solutions X + 2y = 3 2x + 4y =12
System of equations:
[tex]x+2y=3[/tex][tex]2x+4y=12[/tex]To solve the system by graphing, we have to remember that the point in which both graphs meet is the solution of the system.
• Graph of both equations:
As we can see, there is no point in which both meet. Then, this system has no solution.
Answer: none
The formula, = / + , converts temperatures between Celsius and Fahrenheit degrees. What is the temperature in degrees Celsius that is equivalent to 14 degrees FahrenheitA) -10B) -9 C) -8D) -7
Hello there. To answer this question, we need to plug in the value given by the question and solve for C, the temperature in Celsius.
We want to find the equivalent temperature in Celsius to 14 degrees Fahrenheit.
Knowing that F = 9/5C + 32, making F = 14 lead us to:
14 = 9/5C + 32
Subtract 32 on both sides of the equation
9/5C = -18
Multiply both sides of the equation by a factor of 5/9, in order to get:
C = 5/9 (-18) = -10.
This is the equivalent temperature we were looking for.
Using the equation and the ordered-pairs found previously, plot the points on the graph that would best satisfy theequation.y= 2^x
Given the following equation:
[tex]y=x^2[/tex]We will graph the given function using the points that will be written in ordered-pairs.
The given function is a quadratic function with a vertex = (0, 0)
We will graph the points using five points
The vertex and 4 points, 2 points before the vertex and 2 points after the vertex.
So, we will substitute x = -4, -2, 2, 4
[tex]\begin{gathered} x=-4\rightarrow y=16 \\ x=-2\operatorname{\rightarrow}y=4 \\ x=2\operatorname{\rightarrow}y=4 \\ x=4\operatorname{\rightarrow}y=16 \end{gathered}[/tex]So, the points are: (-4, 16), (-2, 4), (0, 0), (2, 4), (4, 16)
The graph using the points will be as follows:
Find the 52nd term.16, 36, 56, 76,…
Answer:
[tex]\text{ a}_{52}\text{ = 1,036}[/tex]Explanation:
Here, we want to find the 52nd term of the sequence
What we have to do here is to check if the sequence is geometric or arithmetic
We can see that:
[tex]\text{ 36-16 = 56-36=76-56 = 20}[/tex]Since the difference between the terms is constant, we can say that the terms have a common difference and that makes the sequence arithmetic
The nth term of an arithmetic sequence can be written as:
[tex]\text{ a}_n\text{ = a +(n-1)d}[/tex]where a is the first term which is given as 16 and d is the common difference which is 20 from the calculation above. n is the term number
We proceed to substitute these values into the formula above
Mathematically, we have this as:
[tex]\begin{gathered} a_{52}\text{ = 16 +(52-1)20} \\ a_{52}\text{ = 16 + (51}\times20) \\ a_{52}\text{ = 16 + 1020 = 1,036} \end{gathered}[/tex]Given parallelogram JKLM, what could the expression 180 - (3x + 8) represents? Explain.
Based on the given figure, you can conclude:
The expression 180 - (3x + 8) represents a supplementary angle to the angle (3x + 8). This angle would be an angles subtended from side KL to a line that is a prolongation of line JK.
3a^2 -3a - 36. solving quadratic by factoring. factor each expression. be sure to check for greatest common factor first.
we have the expression
[tex]3a^2-3a-36[/tex]step 1
Factor 3
[tex]3(a^2-a-12)[/tex]step 2
equate to zero
[tex]3(a^2-a-12)=0[/tex]step 3
Solve
[tex](a^2-a-12)=0[/tex][tex]\begin{gathered} a^2-a=12 \\ (a^2-a+\frac{1}{4}-\frac{1}{4})=12 \\ (a^2-a+\frac{1}{4})=12+\frac{1}{4} \\ (a^2-a+\frac{1}{4})=\frac{49}{4} \end{gathered}[/tex]Rewrite as perfect squares
[tex](a-\frac{1}{2})^2=\frac{49}{4}[/tex]take the square root on both sides
[tex]\begin{gathered} a-\frac{1}{2}=\pm\frac{7}{2} \\ a=\frac{1}{2}\pm\frac{7}{2} \end{gathered}[/tex]the values of a are
a=4 and a=-3
therefore
[tex]3(a^2-a-12)=3(a-4)(a+3)[/tex]the points E,F,G and H all lie on the same line segment, in that order, such that ratio of EG:FG:GH is equal to 4:1:5. If EH=10, find EG
You have that the ratio of EG:FG:GH = 4:1:5.
Moreover, segment EH = 10.
In order to find EG you consider the following ratios:
EG/FG = 4/1
FG/GH = 1/5
Furthermore, EH = EG + FG + GH
Factor the given polynomial by finding the greatest common monomial Factor 6x^3y+9xy^3
Answer:
(3xy)(2x² + 3y²)
Step-by-step explanation:
Hello!
The greatest common factor for the coefficients is 3, as both terms have a coefficient with the greatest factor of 3.
The greatest common factor for the x-terms is x, as both terms has x to a minimum of the first power.
The greatest common factor for the y terms is y as both terms has y to a minimum of the first power.
Factor out 3xy:6x³y + 9xy³3xy(2x²) + 3xy(3y²)(3xy)(2x² + 3y²)The factored form is (3xy)(2x² + 3y²).
Ariana is going to invest $62,000 and leave it in an account for 20 years. Assuming
the interest is compounded continuously, what interest rate, to the nearest tenth of a
percent, would be required in order for Ariana to end up with $233,000?
The rate of interest that Ariana should get in order to end up with a final amount of $233,000 is 0.07%.
What is compound interest and how is it calculated?
The interest that is calculated using both the principal and the interest that has accrued during the previous period is called compound interest. It differs from simple interest in that the principal is not taken into account when determining the interest for the subsequent period with simple interest.
Mathematically, A = P (1 + (R/f))ⁿ ;
where A = amount that the depositor will receive
P = initial amount that the depositor has invested
R = rate of interest offered to the depositor
f = frequency of compounding offered per year
n = number of years.
Given, Amount that Ariana wants to end up receiving = A = $233,00
Principal amount that Ariana can invest = P = $62,000
Frequency of compounding offered per year = f = 1
Number of years = 20
Let the rate of interest offered to the depositor be = R
Following the formula established in the literature, we have:
233000 = 62000(1 + R)²⁰ ⇒ 3.76 = (1 + R)²⁰ ⇒ 1.07 = 1 + R ⇒ R = 0.07%
Thus, the rate of interest that Ariana should get in order to end up with a final amount of $233,000 is 0.07%.
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35×0.37 ..find out the solution and the reasoning behind the solution
Given:
Given that 35×0.37.
Required:
To find the solution and reasoning behind the solution.
Explanation:
Let us manipulate the multiplication.
[tex]35\times0.37=12.95[/tex]As there are two digits after decimal point in 0.37, we put decimal point after two digits from right to left in the product (solution) obtained after multiplication.
Final Answer:
The solution is 12.95.
