The value of a using integration is 2, such that the area of the region bounded by the curve y^2 =4ax and the line x= 4a is 256/3 Sq units
How to determine the value of a?The given parameters are:
Curve, y² = 4ax
Line, x = 4a
Substitute x = 4a in y² = 4ax
y² = 4a * 4a
Take the square root of both sides
y = ±4a
This means that the point of intersection between the line and the curve is (4a, -4a) and (4a, 4a)
Remove the negative interval
So, we have: (4a, 0) and (4a, 4a)
The area is then calculated as
[tex]A =2 \int\limits^{4a}_0 {\sqrt y} \, dx[/tex]
This gives
[tex]A =2 \int\limits^{4a}_0 {\sqrt {4ax}} \, dx[/tex]
Remove the constant
[tex]A = 2\sqrt{4a}\int\limits^{4a}_0 {\sqrt {x}} \, dx[/tex]
Integrate
[tex]A = 2\sqrt{4a} * \frac 23 x^\frac32 |\limits^{4a}_0[/tex]
Evaluate the exponent
[tex]A = 4\sqrt{a} * \frac 23 x^\frac32 |\limits^{4a}_0[/tex]
Expand
[tex]A = 4\sqrt{a} * \frac 23 (4a)^\frac32[/tex]
Rewrite as:
[tex]A = 4 * a^\frac12 * \frac 23 * 4^\frac32 * a^\frac32[/tex]
Evaluate the exponents by the laws of indices
[tex]A = 4 * \frac 23 * 8 * a^2[/tex]
Recall that the area is:
A = 256/3
So, we have:
[tex]\frac{256}3 = 4 * \frac 23 * 8 * a^2[/tex]
Evaluate the quotients
[tex]256 = 4 * 2 * 8 * a^2[/tex]
Divide both sides by 64
4 = a²
Take the square root of both sides
2 = a
Rewrite as:
a = 2
Hence, the value of a is 2
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Which shows a perfect square trinomial?
Answer:
-ax² + bx + c and satisfies the condition b² = 4ac
Step-by-step explanation:
that's my answer
{Correct me if I'm not correct}
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the perimeter of a rectangle is 30cm while it breadth is 5cm find the length of the rectangle
Answer:
10 cm
Step-by-step explanation:
See attached image
Answer:
length = 10 cm
Step-by-step explanation:
Fyi, breadth is a synonym to width.
Since P=30, we must find the length.
l = length
w = width
P=w+w+l+l
30 = 5+5+l+l
30 = 10+2l
-2l = -20
l = 10
Hence, the length is 10 cm.
The area of a circle with a diameter of 240
Answer:
in terms of pi: A = 14,400pi
A = 14,400(3.14) is 45,216
Step-by-step explanation:
Area of a circle is pir^2
Radius = diameter / 2
Radius = 240 / 2 which is 120
A = pi(120)^2
A = 14,400pi
Hi student, let me help you out! :)
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We are asked to find the area of a circle with a diameter of 240.
[tex]\triangle~\fbox{\bf{KEY:}}[/tex]
First of all, we need to find the radius.We can find the radius by dividing the diameter by 2:
[tex]\star~\large\pmb{240\div2=120}[/tex]
The radius is 120. Now it's time to compute the area!
///\\\///\\\///\\\\////\\\///\\\///\\\///\\\///\\\///\\\///\\\///\\\///\\\///\\\////\\\///\\\///\\\///\\\
Computing The Area
The formula for the area is:
[tex]\star~\large\pmb{Area_{(circle)}=\pi r^2}}[/tex]
See, there's the radius, r; we found a minute ago, and it equals 120 cm.
Now, substitute the value of r into the formula:
[tex]\begin{gathered} \sf{\longmapsto{A=\pi (120)^2} \\ \sf{\longmapsto{A=\pi\times14,400} \\ \sf{\longmapsto{A=45,238.9\;cm^2 \end[/tex]
Hope it helps you out! :D
Ask in comments if any queries arise.
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Wendy winholtz 8 year savings certificate pays an annual interest rate of 5.3% at the end of 5 years she cashed out the 6,000 CD and paid a penalty of 12 month simple interest. What penalty did she pay
Answer:
318
Step-by-step explanation:
The rate of 5.3% is an annual rate, and specifies the amount of interest earned in a 12-month period by the principal amount of the CD. The interest amount is found by multiplying the interest rate by the principal amount.
__
penaltyThe amount of the penalty is 5.3% of 6000:
0.053 × 6000 = 318
Wendy paid a penalty of 318 for her early withdrawal.
In ΔABC, m∠A = 2m∠B and m∠B = 3m∠C.
What is m∠C?
A triangle is a three-edged polygon with three vertices. The measure of ∠C is 18°.
What is a triangle?A triangle is a three-edged polygon with three vertices. It is a fundamental form in geometry. The sum of all the angles of a triangle is always equal to 180°.
Let the measure of ∠C be represented by x. Given the measure of ∠A= 2(∠B), while the measure of ∠B=3(∠C). Therefore, the measure of angles can be written as,
∠A = 2(∠B) = 2[3(∠C)] = 6x
∠B = 3x
∠C = x
Now, since the sum of all the angles of a triangle is 180°. Therefore,
∠A + ∠B + ∠C = 180°
6x + 3x + x = 180°
10x = 180°
x = 18°
Hence, the measure of ∠C is 18°.
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Two points A and B whose heights differ by 10 metres are shown 1.2 cm apart horizontally on a map whose scale is 1: 10 000. a) Work out the actual horizontal distance between A and B.
Answer:
120 meters
Step-by-step explanation:
1.2 x 10 000 = 12000 cm = 120 meters
Halp me
Me too dum to know pls halp
Answer:
Coordinate pair for Friday (3, 55). This means the length of the eruption was 3 minutes, and the wait time between the eruptions was 55 minutes.
Paulo and Jason wear smaller shoe sizes than Carmen.
Jason and Carmen scored more runs than Paulo.
Step-by-step explanation:
Paulo wears size 5 and Jason wears size 4. Carmen wears size 10, therefore the players wear smaller shoe sizes than he does
Jason scored 10 runs, and Carmen scored 11. Since Paulo only scored 5 runs, the other players scored more than him
Bobby is having a summer barbecue. He is inviting 7 of his friends. He wants to have enough food for everyone to have two hot dogs. How many hot dogs will bobby need to cook for himself and his friends?
Answer:
16
Step-by-step explanation:
7 friends = 14 hotdogs + bobby (2 hotdogs) = 16 hotdogs
Answer:
16 hot dogs
Step-by-step explanation:
since he has 7 friends + himself, we're going to have a total of 8 persons
and since it's expected for each person to have two hot dogs,
total hotdogs Bobby need to cook= 8×2= 16 hot dogs
i needs help with it please
Answer:
d. ΔMNO
Step-by-step explanation:
In geometry, two figures are congruent if they are the same shape and size (they can also be a mirror image of each other). If they are the same shape but a different size (e,g, one is an enlargement or reduction of the other), then they are similar.
All the given triangles have the same interior angles, so we need to find a triangle with the same side lengths as ΔABC.
The only triangle that has the same side length as ΔABC is ΔMNO:
they both have bases of 8cm.
Therefore, ΔMNO is congruent to ΔABC.
On a coordinate plane, a curved line crosses the x-axis at (negative 1.5, 0), the y-axis at (0, negative 2), and the x-axis at (1.5, 0).
Determine the domain and range of the given function.
The domain is
all real numbers
.
The range is
.
Domain F(x) = All real numbers and the Range F(x) [ -2 , + ∞ ]
What is the domain and range of a function?A domain is the set of values for which the given function is defined.
Range is the set of all values which the given function can output.
The curve points are
P₁ ( -1,5 , 0 ) Q ( 0. -2 ) and P₂ ( 1,5 , 0 )
We can see that P₁ and P₂ are symmetric around the y-axis, and point Q looks like a vertex so we can conclude the curve as
ax² + bx + c = y
The curve is a parabola
when x = 0
y = -2
then c = - 2
The equations we get,
a (-1.5)² + b (-1.5) + c = 0
2.25a - 1.5b - 2 = 0
a (1.5)² + b ( 1.5) + c = 0
2.25a + 1.5b -2 = 0
Adding these two equations
4.5a - 4 = 0
a = 4/4.5
The vertex is Q ( 0 , -2 )
x = - b/2a
0 = - b/ 9
b = 0
Therefore the equation of the function :
ax² + c = 0
(4/4.5) x² - 2 = 0
Therefore, Domain F(x) = All real numbers and the Range F(x) [ -2 , + ∞ ]
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Answer:
In a shorter explanation of what the guy above me is saying
The domain is all real numbers or A
The range is "all real numbers greater than or equal to –2" or B
Step-by-step explanation:
Complete the table of values for the equation, 16x + 4y = 12.
x y
-2
-1
0
1
2
The x-y coordinates for the given equation are: (-2,11),(-1,7),(0,3), (1,-1) and (2,-5).
Linear FunctionA linear function can be represented by a line. The standard form for this equation is: ax+b , for example, y=2x+7. Where:
a= the slope;b=the constant term that represents the y-intercept.The given equation is 16x + 4y = 12. For solving this question, you should replace the given values of x for finding the values of y.
Thus,
For x= -2, the value of y will be:16*(-2)+4y=12
-32+4y=12
4y=12+32
4y=44
y=11
For x= -1, the value of y will be:
16*(-1)+4y=12
-16+4y=124y=12+164y=28y=7For x= 0, the value of y will be:16*(0)+4y=12
0+4y=124y=124y=12y=3For x= 1, the value of y will be:16*(1)+4y=12
16+4y=124y=12-164y=-4y= -1For x= 2, the value of y will be:16*(2)+4y=12
32+4y=124y=12-324y=-20y= -5Read more about the linear equation here:
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Find the value of x. Round to the nearest tenth. X A B 68° 13 x = [?] X Law of Sines; sin A 83° C
Step-by-step explanation:
hope you have a great day
The value of x will be;
⇒ x = 13.9
What is mean by Triangle?A triangle is a three sided polygon, which has three vertices and three angles which has the sum 180 degrees.
Given that;
The triangle is shown in figure.
Now,
Since, We know that;
The sine rule is;
⇒ sin A / a = sin B / b = sin C / c
By given figure, we get;
⇒ sin 83° / x = sin 68° / 13
⇒ sin 83° × 13 / sin 68° = x
⇒ x = 0.99 × 13 / 0.92
⇒ x = 13.9
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Brenda earns $18,200 per year. What’s her semi monthly salary.
Which measure of central tendency can be used in finding the average of your grades in a quarter?
solve the equation
x + 10 = 20
Photo is attached!!!!!! Pleaseeee help!!!
Answer:
1.) ∞
2.) -∞
3.) 0
4.) 0
Step-by-step explanation:
As you approach x = -2 from the left, the "y" values are going to infinity. You never actually get a "y" value that equals x = -2 because of the vertical asymptote.
As you approach x = -2 from the right, the "y" values are going to -infinity. You never actually get a "y" value because there is a vertical asymptote.
As you approach x = 2, both sides converge to y = 0. At this point, there is a smooth, uninterrupted line.
The actual "y" value at x = 2 is 0 because the function goes straight through this point. So, you can just read this point as usual.
Solve for x. Round to the nearest tenth, if necessary.
Answer:
x = 3.4
Step-by-step explanation:
Use the sin formula, due to the fact of the hypotenuse being known, and the variable being across 70.
Sin(70) = x / 3.6
Sin(70) * 3.6 = x
3.3828934 = x
Round to nearest tenth, as question asked
3.4 = x
-- Gage Millar, Algebra 1/2 Tutor (Pending Pre-Calc Tutor)
ƒ(1) = −6
f(2)=-4
f(n) = f(n-2) + f(n-1)
f(3) =
Answer:
-10
Step-by-step explanation:
if we put f(n)=3 , Then
f(3)=f(3-2) + f(3-1)
= f(1) + f(2)
= -6+(-4). [since f(1)= -6 and f(2) = -4)
= -6-4
= -10
I need two question please
What is the range of this data set?
{5, 12, 15, 11, 15, 7, 15, 10, 13}
options:
8
10
12
15
Answer: 10 is the range, option B/ option G
Step-by-step explanation: range is the most subtracted by the least, so organize the data set which makes it (5, 7, 10, 11, 12, 13, 15, 15, 15)
15 - 5 is 10.
Q5. Find each value.
Answer:
[tex]a. {( \sqrt{36} )}^{2} = 36[/tex]
[tex]b. ({ \sqrt{ \frac{25}{81} } })^{2} = \frac{25}{81} = 0.3086[/tex]
[tex]c. {( \sqrt{1.99} )}^{2} = 1.99[/tex]
[tex]d. {( \sqrt{x} })^{2} = x[/tex]
The square exponent is removed with the root .
2 over 5 of a bag of chips is in the cupboard. you eat 3 over 4 of what is left. how ,uch of the whole bag did you eat?
Answer:
3/10
Step-by-step explanation:
So you have 2/5 left in your cupboard. Say the amount of chips originally in your cupboard would be 30. 2/5 of 30 would be 12.
Then multiply 12 by 3/4 so it would be 3/4 of 12.
The result would be 9
9/30 = 3/10
this is complicated explanation
there are much easier explanations tho
A certain city in Washington has a population of 46,400 people.
If the city is 580 sq mi in area, approximately how many people live in each square mile?
Step-by-step explanation:
please can you divide it will give you the answer
Which of the following could be the value of x in the equation x^2 - 21 = 100 Choose all the correct answers
How to solve your problem
x^{2}-21=100
Quadratic formula
Factor
1
Move terms to the left side
x^{2}-21=100
x^{2}-21-100=0
2
Subtract the numbers
x^{2}\textcolor{#C58AF9}{-21}\textcolor{#C58AF9}{-100}=0
x^{2}\textcolor{#C58AF9}{-121}=0
3
Use the quadratic formula
x=\frac{-\textcolor{#F28B82}{b}\pm \sqrt{\textcolor{#F28B82}{b}^{2}-4\textcolor{#C58AF9}{a}\textcolor{#8AB4F8}{c}}}{2\textcolor{#C58AF9}{a}}
Once in standard form, identify a, b, and c from the original equation and plug them into the quadratic formula.
x^{2}-121=0
a=\textcolor{#C58AF9}{1}
b=\textcolor{#F28B82}{0}
c=\textcolor{#8AB4F8}{-121}
x=\frac{-\textcolor{#F28B82}{0}\pm \sqrt{\textcolor{#F28B82}{0}^{2}-4\cdot \textcolor{#C58AF9}{1}(\textcolor{#8AB4F8}{-121})}}{2\cdot \textcolor{#C58AF9}{1}}
4
Simplify
Evaluate the exponent
Multiply the numbers
Add the numbers
Evaluate the square root
Add zero
Multiply the numbers
x=\frac{\pm 22}{2}
5
Separate the equations
To solve for the unknown variable, separate into two equations: one with a plus and the other with a minus.
x=\frac{22}{2}
x=\frac{-22}{2}
6
Solve
Rearrange and isolate the variable to find each solution
x=11
x=-11
make w the subject of the formula p = 2w+7 divided by 12
Answer:
w = (12p - 7) / 2
Step-by-step explanation:
p = (2w + 7) / 12
12p = 2w + 7
2w = 12p - 7
w = (12p - 7) / 2
what equation defines a line that intersects the graph of 6x-2y=10 exactly once?
Any line with a slope different than 3 will intercept the given line only once.
What equation defines a line that intersects the graph?The answer will be any line that is not parallel (nor the same) to the given line.
Remember two lines are parallel if the lines have the same slope and different y-intercept
In this case, the given linear equation is:
6x - 2y = 10
y = 3x - 5
So the slope is 3 and the y-intercept is -5.
Then, any line with a slope different than 3 will intercept this line only once.
An example can be:
y = 314*x
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Which expression represents the perimeter of the triangle?
8x + 2
18x
8x
Answer:
8x
Step-by-step explanation:
We can add all of the side lengths of the triangle to figure out the perimeter. 3x-1+3x+1=6x+2x=8x
19) Florence has a strip of clothing that is 8 feet long. About how many 2 feet strips can she cut from the 8-foot strip?
I really need help with this
Answer:
4
Step-by-step explanation:
8/2 is 4
A rectangle measures 12 m by 14 m. If the width is decreased by 3 m and the length is increased by 4 m, how much would the area change?
Answer:
6 m²
Step-by-step explanation:
Original area :
12 × 14 = 168 m²
New area :
(12-3) × (14+4) =
9 × 18 = 162 m²
168 - 162 = 6
Area would change by 6 m²
Simplify the following: 9 – 2(x - 5) = x + 10
Answer:
x = 3
Step-by-step explanation:
Hello!
We can distribute the parenthesis and solve for x.
Solve9 - 2(x - 5) = x + 109 - 2x + 10 = x + 1019 - 10 = 3x9 = 3xx = 3The value of x is 3.
We can plug it back in to check if it works.
Check9 - 2(x - 5) = x + 109 - 2(3 - 5) = 3 + 109 - 2(-2) = 3 + 109 + 4 = 3 + 1013 = 13Correct!
Answer:
[tex]\huge\boxed{\sf x = 3}[/tex]
Step-by-step explanation:
Given equation:9 - 2(x - 5) = x + 10
Distribute
9 - 2x + 10 = x + 10
Subtract 10 to both sides
9 - 2x = x
Add 2x to both sides
9 = x + 2x
9 = 3x
Divide 3 to both sides
9/3 = 3x/3
3 = x
OR
x = 3[tex]\rule[225]{225}{2}[/tex]