Using the disk method, the volume of the solid is
[tex]\displaystyle \pi \int_3^8 \left(\dfrac1{2x-1}\right)^2 \, dx = \pi \int_3^8 \frac{dx}{(2x-1)^2}[/tex]
Substitute [tex]u = 2x-1[/tex] and [tex]du=2\,dx[/tex].
[tex]\displaystyle \pi \int_3^8 \frac{dx}{(2x-1)^2} = \frac\pi2 \int_5^{15} \frac{du}{u^2} \\\\ ~~~~~~~~ = -\frac\pi2 \frac1u \bigg|_5^{15} \\\\ ~~~~~~~~ = -\frac\pi2 \left(\frac1{15} - \frac15\right) = \boxed{\frac\pi{15}}[/tex]
Pls Help! Geometry
Find the area of the shaded region.
Answer: 112.5 square cm
Step-by-step explanation:
The area of the rectangle is [tex](20)(9)=180[/tex] square cm.
The area of the white triangle is [tex]\frac{1}{2}(9)(20-3-2)=67.5[/tex] square cm.
So, the area of the shaded region is [tex]180-67.5=112.5[/tex] square cm.
what to write it here
Answer:
84 degrees
Step-by-step explanation:
Angle A = 83 degrees
Angle B = x degrees
Angle C = 135 degrees
Angle CDE = 122 degrees
We know that the four inner corners of a quadrilateral should add up to 360 degrees. Two supplementary angles will add up to 180 degrees. Adjacent angles on a straight line will always be supplementary. Knowing this, just solve for <ADC and add that amount to <A and <C. Then, subtract that sum from 360 degrees.
<ADC = 180-122 = 58
58+83+135 = 276
360-276 = 84 degrees
please find the inverse of this
y=log base 4 x^4
Step-by-step explanation:
[tex]y = log_{4}( {x}^{4} ) [/tex]
Swap x and y.
[tex]x = log_{4}( {y}^{4} ) [/tex]
Solve for y.
[tex]x= 4 log_{4}(y) [/tex]
[tex] \frac{1}{4} x = log_{4}(y) [/tex]
[tex]4 {}^{ \frac{1}{4}x } = y[/tex]
So the inverse is
[tex]4 {}^{ \frac{x}{4} } = f {}^{ - 1} (x)[/tex]
Find the Equation of the Perpendicular Line
Instructions: Find the equation of the line through point (-1, 2) and perpendicular to x + 3y = 3.
y =
Answer:
y = 3x +5
Step-by-step explanation:
The equation of a perpendicular line can be formed by swapping the x- and y-coefficients, and negating one of them. The constant in the equation will be chosen to make the equation true at the given point.
Coefficients swappedThe desired equation in the given standard form will be ...
3x -y = c . . . . . . for some new constant c
Note that we have kept the x-coefficient positive, and have negated the y-coefficient.
Constant valueThe new constant will make the equation true at the point (-1, 2):
3(-1) -(2) = c = -5
So, the standard-form equation is ...
3x -y = -5
Slope-intercept formThe answer form suggests you want to solve this for y. Adding y+5 to both sides will give the form you want:
3x -y +(y+5) = -5 +(y+5)
3x +5 = y
y = 3x +5
What is the range of the function shown on the graph above HELP
Answer:
C. 0 ≤ y ≤ 7
Step-by-step explanation:
Range is the interval of the numbers on the y-axis. The numbers go from y-value 1 to y-value 7. Since the circles at the ends are closed, this means 0 and 7 are included in the range, so ≤ is included instead of <.
CIRCLE question is in the picture
Answer:
27π cm² or 84.8 cm²
Explanation:
[tex]\sf Formula \ for \ area \ of \ sector = \dfrac{\theta}{360 } \ x \ \pi (radius)^2[/tex]
Shaded region angle: 360° - 90° = 270°
Given radius: 6 cm
Applying formula:
[tex]\rightarrow \sf \dfrac{270}{360} \ x \ \pi (6)^2[/tex]
[tex]\rightarrow \sf 27\pi[/tex]
[tex]\rightarrow \sf 84.823 \ cm^2[/tex]
[tex]\rightarrow \sf 84.8 \ cm^2 \quad (rounded \ to \ nearest \ tenth)[/tex]
Solve the following equation for X, if Y = 3 and Z = -1. 3XY- 5XZ^2+Y=19
14. If m is a positive integer, then which of the
following is equivalent to 4m + 4m + 4m + 4m?
(A) 4m+1
(B) 4^4m
(C) 4^4m + 1
(D) 4m +4
Hello!
Simplifying :
⇒ 4m + 4m + 4m + 4m
⇒ 4(4m)
None of the above
We have found that:
(X,₁)=(3,-1)
• m = 3
●
Substituting these values in (y - y₁) = m(x-x1), we'll get the equation of the line as:
[tex]\Large\maltese\underline{\textsf{A. What is Asked}}[/tex]
Find the equation of the line, with the info given.
[tex]\Large\maltese\underline{\textsf{This problem has been solved!}}[/tex]
The formula used in this problem is,
[tex]\bf{y-y_1=m(x-x_1)}[/tex]□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
[tex]\bf{y-(-1)=3(x-3)[/tex] | simplify
[tex]\bf{y+1=3(x-3)}[/tex]
[tex]\rule{300}{1.7}[/tex]
[tex]\bf{Result:}[/tex]
[tex]\bf{=y+1=3(x-3)}[/tex]
[tex]\boxed{\bf{aesthetic \not101}}[/tex]
The length of a rectangle is 27 meters and the width is 4 meters. Find the area. Give your answer without units.
Answer:
[tex]108 m^{2}[/tex]
Step-by-step explanation:
[tex]27\times4=108[/tex]
You have $1,300 in savings for retirement. If your investments earn 12% annually, how much will you have in your retirement account in 11 years?
Step-by-step explanation:
Answer :- Amount (after 11 years) = $4522.11
Simplify (38)-2(13⋅38)3(13)4.
The value of (38)-2(13⋅38)3(13)4. is -154090
How to simplify the expression?The expression is given as:
(38)-2(13⋅38)3(13)4.
Rewrite properly as:
(38) - 2 * (13 * 38) * 3 * (13) * 4.
Evaluate the products in the bracket
(38) - 2 * (494) * 3 * 52
Further, expand
38 - 154128
Evaluate the difference
-154090
Hence, the value of (38)-2(13⋅38)3(13)4. is -154090
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7/8 divided by 9/10[tex]7/8 divided by 9/10[/tex]
Step-by-step explanation:
a ) Solution,
[tex] = \frac{7}{8} \div \frac{9}{10} \\ [/tex]
[tex] = \frac{7}{8} \times \frac{10}{9} \\ [/tex]
[tex] = \frac{70}{72} \\ [/tex]
Change into lowest term....
[tex] = \frac{35}{36} \\ [/tex]
Miles is starting a tree farm. His plot of land is triangular with one side 36 feet and the other two sides 30 feet each. The height of this triangle-shaped plot is 24 feet. If each tree needs 8 square feet of space to grow, how many trees can Miles plant?
a. 35 b. 42 c. 54 d. 76
Answer:
54
Step-by-step explanation:
the plot of land is an isosceles triangle, meaning two sides and angles are the same in value.
height (h)= 24
base (b)= 36
length (l)= 30
solution
find the area of triangle using area=half the product of the base and height.
A=1/2(36×24)
A= 216 square feet.
each tree needs 8 square feet
therefore using ratio and proportion,
1 tree= 8 square feet
x trees= 216 square feet
cross multiply and it'll be
216/8=x
therefore x= 54 trees
Solve the system using substitution.
y - 3x = 1
2y - X = 12
([?], [ ?])
Answer:
x = 2 , y = 7
Step-by-step explanation:
Since
y-3x = 1
y = 3x+1 - equation 1
2y-x = 12 - equation 2
Since we are using substitution method,
we will substitute equation 1 into equation 2.
[tex]2(3x + 1) - x = 12 \\ 6x + 2 - x = 12 \\ 5x + 2 = 12 \\ 5x = 12 - 2 \\ 5x = 10 \\ x = \frac{10}{5} \\ = 2[/tex]
Now we substitute x into equation 1 to find y.
[tex]y = 3(2) + 1 \\ = 6 + 1 \\ = 7[/tex]
Therefore x = 2, y = 7.
On Friday, a local hamburger shop sold a combined total of 376 hamburgers and cheeseburgers. The number of cheeseburgers sold was three times the number of hamburgers sold. How many hamburgers were sold on Friday?
Answer:
94 hamburgers
Step-by-step explanation:
Let's solve this question through the use of equations.
Start by defining the variables used.
Let the number of hamburgers and cheeseburgers sold on Friday be h and c respectively.
Form 2 equations using the given information.
Given that the total sold is 376,
h +c= 376 -----(1)
The number of cheeseburgers sold was thrice the number of hamburgers sold.
c= 3h -----(2)
Solving by substitution:
Substitute (2) into (1):
h +3h= 376
Now that we have an equation expressed only in terms of h, we can find the value of h.
4h= 376
Divide both sides by 4:
h= 376 ÷4
h= 94
Thus, 94 hamburgers were sold on Friday.
Write the statement as an equation." Three is subtracted from the reciprocal of a number. Let x be equal to the number .
Answer:
1/x - 3
Step-by-step explanation:
The number is x.
The reciprocal of the number is 1/x.
3 is subtracted from the reciprocal of a number.
1/x - 3
Suppose a graphic designer earns $52,000 and is not self-employed. How much will the designer have to pay in FICA taxes?
The amount that the designer will have to pay in FICA taxes is:$3,978.
FICA taxes
FICA taxes comprises of:
Social Security tax= 6.2%
Medicare tax= 1.45%
Hence:
FICA taxes=($52,000×6.2%)+($52,000×1.45%)
FICA taxes=$3,224+$754
FICA Taxes=$3,978
Therefore the amount that the designer will have to pay in FICA taxes is:$3,978.
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A single die is rolled twice. Find the probability of rolling a 1 the first time and a 4 the second time.
Answer:
1/36
Step-by-step explanation:
There is an 1/6 chance of rolling 1 the first time and 1/6 chance of rolling a 4 the second time. 1/6 * 1/6 = 1/36 chance of rolling a 1 then 4.
What method of matrixes would be used for this question? ( Inverse Matrices, Cramer's Rule, Gaussian Elimination, and Gauss-Jordan Elimination)
May’s restaurant ordered 200 flowers for Mother’s Day. They ordered carnations at $1.50/each, roses at $5.75 each, and daisies at $2.60 each. They ordered mostly carnations, and 20 less roses than daisies. The total order came to $589.50. How many of each type of flower was ordered?
Answer:
(d) Gauss-Jordan Elimination
80 carnations; 50 roses; 70 daisies
Step-by-step explanation:
The given relations can be written as equations, which can be expressed as one matrix equation. Any of the methods listed can be used to solve the matrix equation.
EquationsIf we let c, r, d represent numbers of carnations, roses, and daisies ordered, respectively, then the given relations can be written as ...
c + r + d = 200 . . . . . . 200 flowers were ordered
0c -r +d = 20 . . . . . . . . . . . . . 20 more daisies than roses were ordered
1.50c +5.75r +2.60d = 589.50 . . . . . the total value of the order
Matrix EquationWritten as a matrix equation, it will be of the form ...
AX = B
where A is the square matrix of variable coefficients, X is the column vector of variables, and B is the column vector of equation right-side constants. This is the matrix equation:
[tex]\left[\begin{array}{ccc}1&1&1\\0&-1&1\\1.50&5.75&2.60\end{array}\right] \left[\begin{array}{c}c\\r\\d\end{array}\right] =\left[\begin{array}{c}200\\20\\589.50\end{array}\right][/tex]
Solution MethodsThe mathematical operations required to find the equation solution can be briefly described as ...
Inverse Matrices
The coefficient matrix is inverted and multiplied by the constant column vector:
[tex]X=A^{-1}B[/tex]
The inversion operation requires computation of 10 determinants, of which 9 are of 2×2 matrices. That's a total of about 39 multiplications, 9 divisions, and 20 additions.
Cramer's Rule
Using Cramer's rule requires computation of 4 determinants of 3×3 matrices. The total number of operations comes to about 48 multiplications, 3 divisions, and 20 additions.
Gaussian Elimination
To obtain the upper triangular matrix that results from Gaussian Elimination requires about 11 multiplications, 11 additions, and 2 divisions. This finds the value of one variable, but the others must be found by substitution into the remaining two equations, requiring an additional 3 multiplications and 3 additions.
Gauss-Jordan Elimination
This method starts with an augmented matrix that appends column vector B to the square matrix A. The result of this is shown in the attachment. It is a diagonal matrix with the variable values a direct result of the matrix operations. The calculator's RREF( ) function performs matrix row operations to transform the augmented matrix to this Reduced Row-Echelon Form. About 6 multiplications, 6 additions, and 4 divisions are required.
Clearly, Gauss-Jordan Elimination is the method that requires the least computational work, so it would probably be used for this question.
FlowersThe attachment shows the order to be ...
80 carnations50 roses70 daisies__
Additional comment
The estimates of computational load presented by each of the solution methods are not intended to be exact counts. For this specific problem, some of the operations can be avoided due to the fact that some coefficients are already 1. Also, some computations are not needed simply because they are intended to produce an outcome that is already known. The intention is to give an idea of the relative difficulty of using these different methods.
In some cases, computationally less-efficient methods may be preferred because they are simpler to describe.
what is the percentage of change from 12 to 19
Answer:
(19-12):12*100 =
(19:12-1)*100 =
158.33333333333-100 = 58.33
Help
Note that: The solution must be non zero
Spam/Irrelevant answers will
be reported
Answer:
Step-by-step explanation:
[tex]y'' + \omega^2 y = 0[/tex]
has characteristic equation
[tex]r^2 + \omega^2 = 0[/tex]
with roots at [tex]r = \pm\sqrt{-\omega^2} = \pm|\omega|i[/tex], hence the characteristic solution is
[tex]y = C_1 e^{i|\omega|x} + C_2 e^{-i|\omega|x}[/tex]
or equivalently,
[tex]y = C_1 \cos(|\omega|x) + C_2 \sin(|\omega|x)[/tex]
With the given boundary conditions, we require
[tex]y(0) = 0 \implies C_1 = 0[/tex]
and
[tex]y'(\pi) = 0 \implies -|\omega| C_1 \sin(|\omega|\pi) + |\omega| C_2 \cos(|\omega|\pi) = 0[/tex]
With [tex]C_1=0[/tex], the second condition reduces to
[tex]|\omega| C_2 \cos(|\omega|\pi) = 0[/tex]
Assuming [tex]C_2\neq0[/tex] (because we don't want the trivial solution [tex]y=0[/tex]), it follows that
[tex]\cos(|\omega|\pi) = 0 \implies |\omega|\pi = \pm\dfrac\pi2 + 2n\pi \implies |\omega| = 2n\pm\dfrac12[/tex]
where [tex]n[/tex] is an integer. In order to ensure [tex]|\omega|\ge0[/tex], we must have [tex]n\ge1[/tex] if [tex]|\omega|=2n-\frac12[/tex], or [tex]n\ge0[/tex] if [tex]|\omega|=2n+\frac12[/tex].
Someone please help!
Answer: horizontal
Step-by-step explanation: there going across sideways and not up and down
Instructions: Solve the quadratic equation and simplify your answer(s).
7x2=−12−5x
x=
-5
+
√
,
-5
−
√
14
14
[tex]7x^2 = -12-5x\\\\7x^2 + 5x+12=0\\\\x=\frac{-5 \pm \sqrt{5^{2}-4(7)(12)}}{2(7)}\\\\\boxed{x=\frac{-5 \pm i \sqrt{311}}{14}}[/tex]
If three times a number added to 2 is divided by the number plus 4 the result is eight thirds. Find the number
Answer:
26
Step-by-step explanation:
(3a+2)/(a+4) = 8/3
(3a+2) = (8/3)(a+4)
(3a+2) = a*8/3 + 4*8/3
(3a+2) = 8a/3 + 32/3
3a - 8a/3 = 32/3 - 2
9a/3 - 8a/3 = 32/3 - 6/3
9a - 8a = 32 - 6
a = 26
Check:
(3*26 + 2) / (26+4) = 8/3
(78+2) / 30 = 8/3
80 / 30 = 8/3
TIME REMAINING
59:23
Jessie graphed one of the lines in a system of equations: y = 3 x minus 2. If the system has an infinite number of solutions, which statements are true? Check all that apply.
On a coordinate plane, a line goes through (0, negative 2) and (1, 1).
Any point in the coordinate plane is a solution because it has an infinite number of solutions.
Point (1, 1) is a solution because it is one of the points on the line already graphed.
It is impossible to tell if (–1,–5) is a solution without seeing the other line graphed.
Point (20, 58) is a solution because it results in a true statement when the point values are substituted into the equation of the line.
When the other line in the system is graphed, it will share all points with the line already graphed.
Answer:
a & c
Step-by-step explanation:
Answer:
B) Point (1, 1) is a solution because it is one of the points on the line already graphed.
D) Point (20, 58) is a solution because it results in a true statement when the point values are substituted into the equation of the line.
E) When the other line in the system is graphed, it will share all points with the line already graphed.
Step-by-step explanation:
4. The parents made pies for the school bake sale to pay for the next field trip.
They made 55 apple pies and 45 blueberry pies. How many pies did they make
altogether? Estimate first, then solve.
The total number of pies they made altogether is 100.
How many pies did they make altogether?
In order to determine this value, add the number of blueberry pies with the number of apple pies. Addition is the mathematical operation that is used to determine the sum of two or more numbers.
The total number of pies = 55 + 45 = 100 pies
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What is the greatest common factor of 6, 42, and 18?
Answer:
Step-by-step explanation:
1) List the factors of each number.
Factors of 6 : 1, 2, 3, 6
Factors of 42 : 1, 2, 3, 6, 7, 14, 21, 42
Factors of 18 : 1, 2, 3, 6, 9, 18
2) Find the largest number that is shared by all rows above. This is the GCF.
GCF = 6
For a certain company, the cost for producing x items is 60x+300 and the revenue for selling x items is 100x−0.5x2 .
The profit that the company makes is how much it takes in (revenue) minus how much it spends (cost). In economic models, one typically assumes that a company wants to maximize its profit, or at least wants to make a profit!
Part a: Set up an expression for the profit from producing and selling x items. We assume that the company sells all of the items that it produces. (Hint: it is a quadratic polynomial.)
Part b: Find two values of x that will create a profit of $50 .
The field below accepts a list of numbers or formulas separated by semicolons (e.g. 2;4;6 or x+1;x−1 ). The order of the list does not matter. To enter a−−√ , type sqrt(a).
x=
Preview
Part c: Is it possible for the company to make a profit of $2,500 ?
The answers are
A. The equation for the profit from selling the items is P = 40x-0.5x²-300. b. The values of x are 10 and 70.c. It is not possible to have the profit of 2500How to solve for the expressionWe have the cost function in this question to be
C(x) = 60x+300
We have the function of the revenue to be
Revenue = 100x−0.5x²
A. The formula for revenue function is given
revenue - cost
This is expressed as
= (100x−0.5x²)-60x+300
We have to collect like terms and open the equation above.
This given us:
40x-0.5x²-300B. when profit = 50$
We have p = 40x-0.5x²-300
50 = 40x-0.5x²-300
Multiply the two sides by 10
This gives
500 = 400x - 5x² - 3000
This gives a quadratic equation
5x² - 400x - 3500 = 0
To solve the equation you have to make use of a quadratic calculator.
This gives us the values
x = 10x = 70c. We have P = 40x-0.5x²-300.
at P = 2500
2500 = 40x-0.5x²-300.
Multiply the equation by 10
25000 = 400x - 5x² - 3000
collect like terms
400x - 5x2 - 3000 +28000
400x -5x2 +28000
We have to take the discriminant
-400² - 4*5*28000
= -400000
The discriminant is negative hence it is not possible.
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Choose the letter of the equation for
the graph.
Answer:
a
Step-by-step explanation:
The midline is at y=2.
Eliminate c, d, eAlso, since there is no horizontal shift amongst the remaining options, we know that since y=2 when x=0, this means it must be a sine curve.
This means the answer is a.