Answer:
a=2
Step-by-step explanation:
[tex]area=2\int\limits^a_b {y} \, dx =2\int\limits^a_b {\sqrt{4ax} } \, dx \\=2 \times 2\sqrt{a} \frac{x^{\frac{3}{2} } }{\frac{3}{2} } ~from~~b~to~a\\=\frac{8}{3}\sqrt{a} (a^{\frac{3}{2} } -b^{\frac{3}{2} } )\\=\frac{8}{3} \sqrt{a}( (4a)^{\frac{3}{2} } -0)\\=\frac{8}{3} \sqrt{a} ((4a)\sqrt{4a} -0)\\=\frac{32 a}{3} \times 2a\\=\frac{64}{3} a^2[/tex]
[tex]\frac{64}{3} a^2=\frac{256}{3} \\a^2=\frac{256}{3} \times \frac{3}{64} =4\\a=2 (a > 0)[/tex]
the curve is a right parabola.
here b=0,and a=4a for x
we are finding the area between x-axis and x from 0 t0 4a.
curve is symmetrical about x-axis so we multiply by 2.
If the area of the region bounded by the curve [tex]y^2 =4ax[/tex] and the line [tex]x= 4a[/tex] is [tex]\frac{256}{3}[/tex] Sq units, then the value of [tex]a[/tex] will be [tex]2[/tex] .
What is area of the region bounded by the curve ?An area bounded by two curves is the area under the smaller curve subtracted from the area under the larger curve. This will get you the difference, or the area between the two curves.
Area bounded by the curve [tex]=\int\limits^a_b {x} \, dx[/tex]
We have,
[tex]y^2 =4ax[/tex]
⇒ [tex]y=\sqrt{4ax}[/tex]
[tex]x= 4a[/tex],
Area of the region [tex]=\frac{256}{3}[/tex] Sq units
Now comparing both given equation to get the intersection between points;
[tex]y^2=16a^2[/tex]
[tex]y=4a[/tex]
So,
Area bounded by the curve [tex]= \[ \int_{0}^{4a} y \,dx \][/tex]
[tex]\frac{256}{3} =\[ \int_{0}^{4a} \sqrt{4ax} \,dx \][/tex]
[tex]\frac{256}{3}= \[\sqrt{4a} \int_{0}^{4a} \sqrt{x} \,dx \][/tex]
[tex]\frac{256}{3}= \[2\sqrt{a} \int_{0}^{4a} x^{\frac{1}{2} } \,dx \][/tex]
[tex]\frac{256}{3}= 2\sqrt{a} \left[\begin{array}{ccc}\frac{(x)^{\frac{1}{2}+1 } }{\frac{1}{2}+1 }\end{array}\right] _{0}^{4a}[/tex]
[tex]\frac{256}{3}= 2\sqrt{a} \left[\begin{array}{ccc}\frac{(x)^{\frac{3}{2} } }{\frac{3}{2} }\end{array}\right] _{0}^{4a}[/tex]
[tex]\frac{256}{3}= 2\sqrt{a} *\frac{2}{3} \left[\begin{array}{ccc}(x)^{\frac{3}{2}\end{array}\right] _{0}^{4a}[/tex]
On applying the limits we get;
[tex]\frac{256}{3}= \frac{4}{3} \sqrt{a} \left[\begin{array}{ccc}(4a)^{\frac{3}{2} \end{array}\right][/tex]
[tex]\frac{256}{3}= \frac{4}{3} \sqrt{a} *\sqrt{(4a)^{3} }[/tex]
[tex]\frac{256}{3}= \frac{4}{3} \sqrt{a} * 8 *a^{2} \sqrt{a}[/tex]
[tex]\frac{256}{3}= \frac{4}{3} * 8 *a^{3}[/tex]
⇒ [tex]a^{3} =8[/tex]
[tex]a=2[/tex]
Hence, we can say that if the area of the region bounded by the curve [tex]y^2 =4ax[/tex] and the line [tex]x= 4a[/tex] is [tex]\frac{256}{3}[/tex] Sq units, then the value of [tex]a[/tex] will be [tex]2[/tex] .
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Consider the linear system:
[tex]\overrightarrow y'=\begin{bmatrix}-6 & -4 \\12 & 8\end{bmatrix}\overrightarrow y[/tex]
a. Find the eigenvalues and eigenvectors for the coefficient matrix.
b. For each eigenpair in the previous part, form a solution of [tex]\overrightarrow y' = A\overrightarrow y[/tex]. Use [tex]t[/tex] as the independent variable in your answers.
c. Does the set of solutions you found form a fundamental set (i.e., linearly independent set) of solutions?
a. If A is the coefficient matrix, solve det(A - λI) = 0 for the eigenvalues λ :
[tex]\det\begin{bmatrix}-6-\lambda & -4 \\ 12 & 8-\lambda\end{bmatrix} = (-6-\lambda)(8-\lambda)+48 = 0 \implies \lambda(\lambda-2)=0[/tex]
[tex]\implies \lambda = 0, \lambda = 2[/tex]
Let v = [v₁, v₂]ᵀ be the eigenvector corresponding to λ. Solve Av = λv for v :
[tex]\lambda=0 \implies \begin{bmatrix}-6&-4\\12&8\end{bmatrix}\begin{bmatrix}v_1\\v_2\end{bmatrix} = \begin{bmatrix}0\\0\end{bmatrix} \implies 3v_1 + 2v_2 = 0[/tex]
If we pick v₂ = -3, then v₁ = 2, so [2, -3]ᵀ is the eigenvector for λ = 0.
[tex]\lambda = 2 \implies \begin{bmatrix}-8&-4\\12&6\end{bmatrix}\begin{bmatrix}v_1\\v_2\end{bmatrix} = \begin{bmatrix}0\\0\end{bmatrix} \implies 2v_1 + v_2 = 0[/tex]
Let v₁ = 1, so v₂ = -2.
b. λ = 0 and v = [2, -3]ᵀ contributes a constant solution,
[tex]\vec y_1 = e^{\lambda t} v = \begin{bmatrix}2\\-3\end{bmatrix}[/tex]
while λ = 2 and v = [1, -2]ᵀ contribute a solution of the form
[tex]\vec y_2 = e^{\lambda t} v = e^{2t} \begin{bmatrix}1\\-2\end{bmatrix}[/tex]
c. Yes; compute the Wronskian of the two fundamental solutions:
[tex]W(1, e^{2t}) = \det\begin{bmatrix}1 & e^{2t} \\ 0 & 2e^{2t}\end{bmatrix} = 2e^{2t} \neq 0[/tex]
The Wronskian is non-zero, so the solutions are independent.
I don’t quite understand this problem could someone help me please
Answer:
[tex]\frac{7\sqrt{65}}{65}[/tex]
Step-by-step explanation:
Cosine is the ratio of the side adjacent to the angle and the right triangle hypotenuse.
[tex]cos[/tex] B = [tex]\frac{7}{\sqrt{65}}[/tex] = [tex]\frac{7\sqrt{65}}{65}[/tex]
FGH and HGJ form a linear pair. Find the measurement of the anglesif FGH=11x and HGJ+(6x-7)
The angles are 121 and 59 degrees, respectively
How to determine the angles?The given parameters are:
FGH = 11x
HGJ = 6x - 7
Linear pair angles add up to 180.
So, we have:
11x + 6x - 7 = 180
Evaluate the like terms
17x = 187
Divide both sides by 17
x = 11
Substitute x = 11 in FGH = 11x and HGJ = 6x - 7
FGH = 11 * 11=121
HGJ = 6 * 11 - 7 = 59
Hence, the angles are 121 and 59 degrees, respectively
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Round your answer to the nearest tenth if necessary. Use 3.14 for.
A pan for baking bread is shaped like half a cylinder. It is 16 inches long and 4.5 inches in diameter.
What is the volume of uncooked dough that would fill this pan?
4.5 in.
16 in.
The volume is approximately
in³.
just a little help
Answer:
127.17 in³
Step-by-step explanation:
V = (1/2)πr²h
(1/2)(3.14)(2.25 in)²(16 in)
(1/2)(3.14)(5.0625 in²)(16 in) = 127.17 in³
What is the volume of a cylinder with base radius 2 and height 9?
Answer:
V =113.09734
Step-by-step explanation:
[tex]V=\pi r^{2}[/tex]
[tex]= 2^{2}[/tex] x 9= 113.09734
(which you can technically round to 1)
=113.1
How is the product of 3 and –2 shown using integer tiles?
3 positive tiles and 2 negative tiles.
3 negative tiles and 2 positive tiles.
3 sets of 2 negative tiles.
3 sets of 2 positive tiles.
A triangle has vertices at (2, 3), (-4, 5), and (-3, 4). What are the coordinates of the vertices
of the image after the translation (x, y) (x + 1, y - 3)?
[tex](2,3) \longrightarrow (2+1, 3-3)=\boxed{(3,0)}\\(-4,5) \longrightarrow (-4+1, 5-3)=\boxed{(-3, 2)}\\(-3,4) \longrightarrow (-3+1, 4-3)=\boxed{(-2, 1)}[/tex]
let a and b be roots of x² - 4x + 2 = 0. find the value of a/b² +b/a²
Answer:
[tex]\dfrac a{b^2} + \dfrac b{a^2} = 10[/tex]
Step-by-step explanation:
[tex]\text{Given that, the roots are a,b and } ~ x^2 -4x+2 = 0\\\\\text{So,}\\\\a+b = -\dfrac{-4}1 = 4\\\\ab = \dfrac 21 = 2\\\\\text{Now,}\\\\~~~~~\dfrac a{b^2} + \dfrac b{a^2}\\\\\\=\dfrac{a^3 +b^3}{a^2b^2}\\\\\\=\dfrac{(a+b)^3 -3ab(a+b)}{(ab)^2}\\\\\\=\dfrac{4^3 -3(2)(4)}{2^2}\\\\\\=\dfrac{64-24}{4}\\\\\\=\dfrac{40}{4}\\\\\\=10[/tex]
Answer:
[tex]\dfrac{a}{b^2}+\dfrac{b}{a^2}=10[/tex]
Step-by-step explanation:
Given equation: [tex]x^2-4x+2=0[/tex]
The roots of the given quadratic equation are the values of x when [tex]y=0[/tex].
To find the roots, use the quadratic formula:
[tex]x=\dfrac{-b \pm \sqrt{b^2-4ac} }{2a}\quad\textsf{when }\:ax^2+bx+c=0[/tex]
Therefore:
[tex]a=1, \quad b=-4, \quad c=2[/tex]
[tex]\begin{aligned}\implies x & =\dfrac{-(-4) \pm \sqrt{(-4)^2-4(1)(2)}}{2(1)}\\& =\dfrac{4 \pm \sqrt{8}}{2}\\& =\dfrac{4 \pm 2\sqrt{2}}{2}\\& =2 \pm \sqrt{2}\end{aligned}[/tex]
[tex]\textsf{Let }a=2+\sqrt{2}[/tex]
[tex]\textsf{Let }b=2-\sqrt{2}[/tex]
Therefore:
[tex]\begin{aligned}\implies \dfrac{a}{b^2}+\dfrac{b}{a^2} & = \dfrac{2+\sqrt{2}}{(2-\sqrt{2})^2}+\dfrac{2-\sqrt{2}}{(2+\sqrt{2})^2}\\\\& = \dfrac{2+\sqrt{2}}{6-4\sqrt{2}}+\dfrac{2-\sqrt{2}}{6+4\sqrt{2}}\\\\& = \dfrac{(2+\sqrt{2})(6+4\sqrt{2})+(2-\sqrt{2})(6-4\sqrt{2})}{(6-4\sqrt{2})(6+4\sqrt{2})}\\\\& = \dfrac{12+8\sqrt{2}+6\sqrt{2}+8+12-8\sqrt{2}-6\sqrt{2}+8}{36+24\sqrt{2}-24\sqrt{2}-32}\\\\& = \dfrac{40}{4}\\\\& = 10\end{aligned}[/tex]
In function notation, f(x)is used instead of the letter ___ to represent the __________ variable.
In function notation, f(x) is used instead of the letter y to represent the output variable.
How to complete the blanks?A function is represented as:
f(x)
The above means that:
The function of x
As a general rule, the function can be rewritten using the letter y (i.e. the output variable)
Hence, the words that complete the blanks are y and output
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daniel wants to build a sandbox that has a perimeter or 20.5 ft. the length is 5.36 ft. what is the width of the sandbox?
Answer:
20.5 - 5.36
= 15.14/2 (because we are using perimeter, so there are 2 equal lengths and 2 equal widths)
=7.57, therefore the width of the sandbox is 7.57 ft.
Answer:
4.89 ft
Step-by-step explanation:
Perimeter of a rectangle is given by the formula :
P = 2L + 2W
So now we solve for W :
P = 2(L+W)
P/2 = L+W
P/2 - L = W
Now we substitute our P and L into the rearranged formula :
20.5 / 2 - 5.36 = W
10.25 - 5.36 = W
W = 4.89 ft
Hope this helped and brainliest please
Click on the pic there will be the question (very easy I’m just lazy lol)
Answer: ......... 14.85
Step-by-step explanation:
4.95 × 3
...
14.85
bros lazy lol
Pencils cost $0.05. Notebooks cost $0.30. Henry spent $1.40. How many of each did he buy if he bought the same number of pencils and notebooks? A. 3 B. 4 C. 6 D. 8
Answer:
b) 4
Step-by-step explanation:
4 times $0.05 = $0.20
4 times $0.30 = $1.20
then you add the both totals together
0.20 + 1.20= $1.40
Which equation matches the graph of the greatest integer function given
below?
Answer: is y=[x]-2
I used this app to help me get the answer too lol
Which choice shows 19 + 32 + 11 rewritten correctly using the commutative property and then simplified correctly?
19+43 = 62
19+11+32= 30+32 = 62
11+9+10+32 20+32 = 52
19+11+32 = 30 +43 = 73
The choice shows 19 + 32 + 11 rewritten correctly using the commutative property and then simplified correctly is option B; 19+11+32= 30+32 = 62.
What is the commutative property of addition?The commutative property of addition says that it doesn't matter how we add two numbers, the result of the addition would be same.
For two numbers x and y, we have:
x + y = y + x
WE have given
19 + 32 + 11
The sum would be 62.
We know that by commutative property of addition,
For two numbers x and y,
x + y = y + x
Thus, if we take 3 numbers as a,b and c, then:
c + a + b = c + b + a
Similarly,
19 + 32 + 11 = 19+11+32
= 30+32
= 62
Therefore, the option B is the correct answer.
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Translate (13, 4) to the right 5 and up 2
Answer:
(18,6)
Step-by-step explanation:
add 5 units to X and 2 to Y
please help i dont know
√324 + 6 √64
what is the answer to this question.
Answer:
66
Step-by-step explanation:
[tex]\sqrt{324} + 6 \sqrt{64}\\\\=\sqrt{18^2} +6 \sqrt{8^2}\\\\=18+6(8)\\\\=18+48\\\\=66[/tex]
Answer:
hmm
Step-by-step explanation:
not enough words to know u steal my points I steal yours kid
The table gives information about the height of some trees draw a histogram for the information of some trees
2.3+2.2y uneed to pus the 2 on the 3 and answer is uten
The linear equation y = 25x describes how far from home Gary is as he drives from Montreal to Miami. Let x represent the number of hours and y represent the number of miles. How far from home is Gary in 12 hours? Graph the equation and tell whether it is linear.The linear equation y = 25x describes how far from home Gary is as he drives from Montreal to Miami. Let × represent the number of hours and y represent the number of miles. How far from home is Gary in 12 hours? Graph the equation and tell whether it is linear.
The distance that Greg is from home after 12 hours is given as follows:
300 miles.
How to calculate the numeric value of a function or of an expression?To calculate the numeric value of a function or of an expression, we substitute each instance of any variable or unknown on the function by the value at which we want to find the numeric value of the function or of the expression presented in the context of a problem.
The function for this problem is given as follows:
y = 25x.
Hence the distance after 12 hours is given as follows:
y = 25 x 12
y = 300 miles.
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Please help Urgently and need working out!
A line intersects the points (-5,1) and (-2,7). m=2 Write an equation in point slope form using the point (-5,1) y-[?] = ___(x-___)
[tex](\stackrel{x_1}{-5}~,~\stackrel{y_1}{1})\qquad (\stackrel{x_2}{-2}~,~\stackrel{y_2}{7}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{7}-\stackrel{y1}{1}}}{\underset{run} {\underset{x_2}{-2}-\underset{x_1}{(-5)}}} \implies \cfrac{7 -1}{-2 +5}\implies \cfrac{6}{3}\implies 2[/tex]
[tex]\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{1}=\stackrel{m}{2}(x-\stackrel{x_1}{(-5)})\implies y-1=2(x+5)[/tex]
6 children were jumping rope.
5 children joined them.
2 children left because the line was too long.
How many children were jumping rope then
Answer:
9 children were jumping then.
Step-by-step explanation:
6+5 equals to 11 - 2 of the children left would equal 9 children left.
with full explanation from the internet like before
1/(x-5)+3/(x+2)=4
Solution :
[tex]x = \frac{4 + \sqrt{43} }{2} .x = 4 + \frac{4 - \sqrt{43} }{2} [/tex]
Step-by-step explanation:[tex] \frac{1}{(x - 5)} + \frac{3}{(x - 2)} - 4[/tex]
1. Multiply by LCM[tex]x = 2 + 3( x - 5) = 4 - (x - 5)(x + 2)[/tex]
2. Solve[tex]x = 2 + 3(x - 5) = 4 (x - 5)(x + 2)[/tex][tex]x = \frac{4 + \sqrt{43} }{2} .x = 4 + \frac{4 - \sqrt{43} }{2} [/tex]3.Verify SolutionsFind undefined (singularity) points : x=5,x=–2[tex]x = \frac{4 + \sqrt{43} }{2} .x = 4 + \frac{4 - \sqrt{43} }{2} [/tex]
[tex]\\ \rm\Rrightarrow \dfrac{1}{x-5}+\dfrac{3}{x+2}=4[/tex]
[tex]\\ \rm\Rrightarrow \dfrac{x+2+3x-15}{(x-5)(x+2)}=4[/tex]
[tex]\\ \rm\Rrightarrow 4x-13=4(x-5)(x+2)[/tex]
[tex]\\ \rm\Rrightarrow 4x-13=4(x(x+2)-5(x+2))[/tex]
[tex]\\ \rm\Rrightarrow 4x-13=4(x^2+2x-5x-10)[/tex]
[tex]\\ \rm\Rrightarrow 4x-13=4(x^2-3x-10)[/tex]
[tex]\\ \rm\Rrightarrow 4x-13=4x^2-12x-40[/tex]
[tex]\\ \rm\Rrightarrow 4x^2-16x-53=0[/tex]
On solving we get
[tex]\\ \rm\Rrightarrow x=2\pm\dfrac{69}{2}[/tex]
Determinar cuales de las siguientes frases son proposiciones
a) 3+2 = 0 c) ¡Hola!
b) x + 1 = 4 d) Yo estudio
The exercise is designed to test the students knowledge of prepositions. The correct answer thus is (Option D) Yo estudio (which translates) "I study".
What is the explanation for the answer above?To understand the answer, you need to know what a preposition is. A preposition, in simple terms, is a word or group of words that comes before a noun.
The function of a preposition (much like an adjective) is to give clarity to the noun that it precedes.
Let us complete the Preposition phrase to give meaning to it:
"I study Mathematics". Where;
"Mathematics" is the noun;
"I Study" is the prepositional phrase. Hence the correction answer to the question above is Option D.
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answer the problem please
Answer: Point C is (-2, -2), which is the third option.
Step-by-step explanation:
The midpoint formula is [tex]m=(\frac{x_{1}+x_{2} }{2},\frac{y_{1}+y_{2} }{2})[/tex]
All we have to do is plug in the numbers and solve the fractions from there.
[tex]m=(\frac{-10+6}{2},\frac{-6+2}{2} )[/tex]
[tex](\frac{-4}{2},\frac{-4}{2} )[/tex]
[tex]m=(-2, -2)[/tex]
I hope this helps!!
7x-5=30
I need this explained for me please
Step-by-step explanation:
7x-5=30
Firstly -5 will cross the equality sign and become +5. so we'll have
7x=30+5=35
7x=35 we'll divide both sides by 7
giving u
7x/7=35/7
=x=5
therefore x=5.
CONFIRMATION
7*5-5=30
PLS MARK ME AS BRAINLIEST
Answer:
x = 5.Step-by-step explanation:
Solve for X:
7x - 5 = 30= 7 * x= 7(5)= 35 - 5= 30x = 5.Hence, answer is x = 5.If f(x) = 2x + 3, what is f(–2)?
Substitute -2 for x
[tex]f( - 2) = 2( - 2) + 3 \\ y = - 4 + 3 \\ y = - 1[/tex]
Hope it helps
Please give brainliest
Answer:
-1
Step-by-step explanation:
Hi student! Let me help you out on this question.
_____________________
To find the value of f(-2), we need to stick in -2 for x.
[tex]\mathsf{f(-2)=2\cdot(-2)+3}[/tex]. Multiply first.
[tex]\mathrm{f(-2)=-4+3}[/tex]. Now simplify completely.
[tex]\mathsf{f(-2)=-1}[/tex]. Which is our final answer.
Hope that this helped you out! have a good day ahead.
Best Wishes!
[tex]\star\bigstar\underline{\underline{\overline{\overline{\textsf{Reach far. Aim high. Dream big.}}}}}\bigstar\star[/tex]
◆◈-Greetings!-◆◈
__________________
log x + log (2x-1) = log 6
Answer:
x = 2
Step-by-step explanation:
logx + log(2x - 1) = log 6
Apply log rules,
x(2x - 1) = 6
2x^2 - 1x = 6
x = 2 (true) or x = -3/2 (False)
solve for y :
[tex]\longrightarrow \: \bold{3 + y = 18}[/tex]
ty! ~
Answer:
y = 15
Step-by-step explanation:
Given :
3 + y = 18
This a simple algebraic equation.
Subtract 3 from both sides :
⇒ 3 + y - 3 = 18 - 3
⇒ y = 15
Answer:
y = 15
Step-by-step explanation:
Given equation:
3+y=18To Find:
Value of ySolution:
We can rewrite this equation as:
y+3 = 8[This'll ain't change the answer]
Now we could solve on a easy way.
STEPS:
Transpose +3 to the RHS, make sure to change it's sign from “+” to “-”.
=> y = 18-3
Subtract the integers which's on the RHS:
=> y = 15
Hence,the value of y will be 15.
[tex] \rule{225pt}{2pt}[/tex]
[tex]\frac{1}{2} (5x - 9 ) = 2 (\frac{1}{3} + 6 )[/tex]
Answer: 103/15
Step-by-step explanation:
We can simplify the right-hand side to be [tex]2 \left(\frac{1}{3}+6 \right)=2 \left(\frac{19}{3} \right)=\frac{38}{3}[/tex].
This means we need to solve:
[tex]\frac{1}{2}(5x-9)=\frac{38}{3}\\5x-9=\frac{76}{3}\\5x=\frac{103}{3}\\x=\boxed{\frac{103}{15}}[/tex]