Answer:
The answer is [tex]125.64[/tex]
Step-by-step explanation:
hope this helps :)
The portion of the parabola y²=4ax above the x-axis, where is form 0 to h is revolved about the x-axis. Show that the surface area generated is
A=8/3π√a[(h+a)³/²-a³/2]
Use the result to find the value of h if the parabola y²=36x when revolved about the x-axis is to have surface area 1000.
Answer:
See below for Part A.
Part B)
[tex]\displaystyle h=\Big(\frac{125}{\pi}+27\Big)^\frac{2}{3}-9\approx7.4614[/tex]
Step-by-step explanation:
Part A)
The parabola given by the equation:
[tex]y^2=4ax[/tex]
From 0 to h is revolved about the x-axis.
We can take the principal square root of both sides to acquire our function:
[tex]y=f(x)=\sqrt{4ax}[/tex]
Please refer to the attachment below for the sketch.
The area of a surface of revolution is given by:
[tex]\displaystyle S=2\pi\int_{a}^{b}r(x)\sqrt{1+\big[f^\prime(x)]^2} \,dx[/tex]
Where r(x) is the distance between f and the axis of revolution.
From the sketch, we can see that the distance between f and the AoR is simply our equation y. Hence:
[tex]r(x)=y(x)=\sqrt{4ax}[/tex]
Now, we will need to find f’(x). We know that:
[tex]f(x)=\sqrt{4ax}[/tex]
Then by the chain rule, f’(x) is:
[tex]\displaystyle f^\prime(x)=\frac{1}{2\sqrt{4ax}}\cdot4a=\frac{2a}{\sqrt{4ax}}[/tex]
For our limits of integration, we are going from 0 to h.
Hence, our integral becomes:
[tex]\displaystyle S=2\pi\int_{0}^{h}(\sqrt{4ax})\sqrt{1+\Big(\frac{2a}{\sqrt{4ax}}\Big)^2}\, dx[/tex]
Simplify:
[tex]\displaystyle S=2\pi\int_{0}^{h}\sqrt{4ax}\Big(\sqrt{1+\frac{4a^2}{4ax}}\Big)\,dx[/tex]
Combine roots;
[tex]\displaystyle S=2\pi\int_{0}^{h}\sqrt{4ax\Big(1+\frac{4a^2}{4ax}\Big)}\,dx[/tex]
Simplify:
[tex]\displaystyle S=2\pi\int_{0}^{h}\sqrt{4ax+4a^2}\, dx[/tex]
Integrate. We can consider using u-substitution. We will let:
[tex]u=4ax+4a^2\text{ then } du=4a\, dx[/tex]
We also need to change our limits of integration. So:
[tex]u=4a(0)+4a^2=4a^2\text{ and } \\ u=4a(h)+4a^2=4ah+4a^2[/tex]
Hence, our new integral is:
[tex]\displaystyle S=2\pi\int_{4a^2}^{4ah+4a^2}\sqrt{u}\, \Big(\frac{1}{4a}\Big)du[/tex]
Simplify and integrate:
[tex]\displaystyle S=\frac{\pi}{2a}\Big[\,\frac{2}{3}u^{\frac{3}{2}}\Big|^{4ah+4a^2}_{4a^2}\Big][/tex]
Simplify:
[tex]\displaystyle S=\frac{\pi}{3a}\Big[\, u^\frac{3}{2}\Big|^{4ah+4a^2}_{4a^2}\Big][/tex]
FTC:
[tex]\displaystyle S=\frac{\pi}{3a}\Big[(4ah+4a^2)^\frac{3}{2}-(4a^2)^\frac{3}{2}\Big][/tex]
Simplify each term. For the first term, we have:
[tex]\displaystyle (4ah+4a^2)^\frac{3}{2}[/tex]
We can factor out the 4a:
[tex]\displaystyle =(4a)^\frac{3}{2}(h+a)^\frac{3}{2}[/tex]
Simplify:
[tex]\displaystyle =8a^\frac{3}{2}(h+a)^\frac{3}{2}[/tex]
For the second term, we have:
[tex]\displaystyle (4a^2)^\frac{3}{2}[/tex]
Simplify:
[tex]\displaystyle =(2a)^3[/tex]
Hence:
[tex]\displaystyle =8a^3[/tex]
Thus, our equation becomes:
[tex]\displaystyle S=\frac{\pi}{3a}\Big[8a^\frac{3}{2}(h+a)^\frac{3}{2}-8a^3\Big][/tex]
We can factor out an 8a^(3/2). Hence:
[tex]\displaystyle S=\frac{\pi}{3a}(8a^\frac{3}{2})\Big[(h+a)^\frac{3}{2}-a^\frac{3}{2}\Big][/tex]
Simplify:
[tex]\displaystyle S=\frac{8\pi}{3}\sqrt{a}\Big[(h+a)^\frac{3}{2}-a^\frac{3}{2}\Big][/tex]
Hence, we have verified the surface area generated by the function.
Part B)
We have:
[tex]y^2=36x[/tex]
We can rewrite this as:
[tex]y^2=4(9)x[/tex]
Hence, a=9.
The surface area is 1000. So, S=1000.
Therefore, with our equation:
[tex]\displaystyle S=\frac{8\pi}{3}\sqrt{a}\Big[(h+a)^\frac{3}{2}-a^\frac{3}{2}\Big][/tex]
We can write:
[tex]\displaystyle 1000=\frac{8\pi}{3}\sqrt{9}\Big[(h+9)^\frac{3}{2}-9^\frac{3}{2}\Big][/tex]
Solve for h. Simplify:
[tex]\displaystyle 1000=8\pi\Big[(h+9)^\frac{3}{2}-27\Big][/tex]
Divide both sides by 8π:
[tex]\displaystyle \frac{125}{\pi}=(h+9)^\frac{3}{2}-27[/tex]
Isolate term:
[tex]\displaystyle \frac{125}{\pi}+27=(h+9)^\frac{3}{2}[/tex]
Raise both sides to 2/3:
[tex]\displaystyle \Big(\frac{125}{\pi}+27\Big)^\frac{2}{3}=h+9[/tex]
Hence, the value of h is:
[tex]\displaystyle h=\Big(\frac{125}{\pi}+27\Big)^\frac{2}{3}-9\approx7.4614[/tex]
You seem to have left out that 0 ≤ x ≤ h.
From y² = 4ax, we get that the top half of the parabola (the part that lies in the first quadrant above the x-axis) is given by y = √(4ax) = 2√(ax). Then the area of the surface obtained by revolving this curve between x = 0 and x = h about the x-axis is
[tex]2\pi\displaystyle\int_0^h y(x) \sqrt{1+\left(\frac{\mathrm dy(x)}{\mathrm dx}\right)^2}\,\mathrm dx[/tex]
We have
y(x) = 2√(ax) → y'(x) = 2 • a/(2√(ax)) = √(a/x)
so the integral is
[tex]4\sqrt a\pi\displaystyle\int_0^h \sqrt x \sqrt{1+\frac ax}\,\mathrm dx[/tex]
[tex]=\displaystyle4\sqrt a\pi\int_0^h (x+a)^{\frac12}\,\mathrm dx[/tex]
[tex]=4\sqrt a\pi\left[\dfrac23(x+a)^{\frac32}\right]_0^h[/tex]
[tex]=\dfrac{8\pi\sqrt a}3\left((h+a)^{\frac32}-a^{\frac32}\right)[/tex]
Now, if y² = 36x, then a = 9. So if the area is 1000, solve for h :
[tex]1000=8\pi\left((h+9)^{\frac32}-27\right)[/tex]
[tex]\dfrac{125}\pi=(h+9)^{\frac32}-27[/tex]
[tex]\dfrac{125+27\pi}\pi=(h+9)^{\frac32}[/tex]
[tex]\left(\dfrac{125+27\pi}\pi\right)^{\frac23}=h+9[/tex]
[tex]\boxed{h=\left(\dfrac{125+27\pi}\pi\right)^{\frac23}-9}[/tex]
Drag the missing word into place
swhen they enter the balls throw
Or you run real quick to get home
Solar
ENGTE
Kinetic
Answer:
kinetic energy
Step-by-step explanation:
When a person runs, their body must convert potential energy into kinetic energy.
3. To make purple frosting, you start with white frosting.
Then you add 2 drops of red food coloring and 3 drops of
blue food coloring. If 4 red drops come out with the first
squeeze, how many blue drops should you add?
==================================================
Explanation:
We basically mix red and blue to get purple. The ratio needed is 2 parts red, and 3 parts blue. We can shorten this by writing 2:3
Note how we can double both parts of the ratio to get 4:6, showing that 4 drops of red correspond to 6 drops of blue, which is the answer we want.
------------------
Another approach:
x = some positive real number
(2 parts red)/(3 parts blue) = (4 parts red)/(x parts blue)
2/3 = 4/x
2x = 3*4 .... cross multiply
2x = 12
x = 12/2 .... divide both sides by 2
x = 6
So 6 drops of blue go with 4 drops of red.
A rhombus is not a
a. polygon. c. trapezoid.
b. parallelogram. d. quadrilateral.
Answer:
polygon
Step-by-step explanation:
A polygon is any 2-dimensional shape formed with straight lines. Triangles, quadrilaterals, pentagons, and hexagons are all examples of polygons. A rhombus has 4 sides.
Hope this helps:) Have a great day!
Bags of pretzels are sampled to ensure proper weight. The overall average for the samples is 9 ounces. Each sample contains 25 bags. The standard deviation is estimated to be 3 ounces. The upper control chart limit (for 99.7% confidence) for the average would be ________ ounces.
Answer:
The value is [tex]UCL = 10.8[/tex]
Step-by-step explanation:
From the question we are told that
The sample mean is [tex]\= x = 9 \ ounce[/tex]
The sample size is n = 25
The standard deviation is [tex]\sigma = 3 \ ounce[/tex]
Given that the sample size is not large enough i.e n< 30 we will make use of the student t distribution table
From the question we are told the confidence level is 99.7% , hence the level of significance is
[tex]\alpha = (100 -99.7 ) \%[/tex]
=> [tex]\alpha = 0.003[/tex]
Generally the degree of freedom is [tex]df = n- 1[/tex]
=> [tex]df = 25 - 1[/tex]
=> [tex]df = 24[/tex]
Generally from the student t distribution table the critical value of [tex]\frac{\alpha }{2}[/tex] at a degree of freedom of [tex]df = 24[/tex] is
[tex]t_{\frac{\alpha }{2} , 24 } = 3.0 [/tex]
Generally the margin of error is mathematically represented as
[tex]E = t_{\frac{\alpha }{2} , 24} * \frac{\sigma }{\sqrt{n} }[/tex]
=> [tex]E = 3.0 * \frac{3 }{\sqrt{25} }[/tex]
=> [tex]E =1.8 [/tex]
Gnerally the upper control chart limit for 99.7% confidence is mathematically represented as
[tex]UCL = \= x + E[/tex]
=> [tex]UCL = 9 + 1.8[/tex]
=> [tex]UCL = 10.8[/tex]
How much simple interest does $5,000 earn in 4 years at an interest rate of 5%. Round to
the nearest cent.
Answer:
1000
Step-by-step explanation:
What is the length of x?
Answer:
25
Step-by-step explanation:
a2+b2=c2
3^2+4^2=c^2
9+16=25
25^2=625
Sqrt of 625 = 25
Answer:
x = 5
Step-by-step explanation:
Using Pythagoras' identity in the right triangle.
The square on the hypotenuse is equal to the sum of the squares on the other 2 sides, that is
x² = 4² + 3² = 16 + 9 = 25 ( take the square root of both sides )
x = [tex]\sqrt{25}[/tex] = 5
Please help don’t pass this question please help me out !
Answer: I think it is (D)
Step-by-step explanation: Its hard to explain but I hope this helps anyways!
Answer:
I think its D to so I would go with that
What is the zero of 4x - 3y = -24?
Rose and Angel are caterers. Both Rose and Angel charge a flat fee plus a fee per guest. Rose charges a flat fee of $50 plus $12 per guest. The table shows how much Angel charges for different numbers of guests. Angel's Catering Charges Number of Guests Charge ($) 10 $180 15 $255 25 $405 30 $480 Part A Who charges the greater flat fee? O A. Rapse O B. Angel Part B Vhat is the difference in total charges between Angel and Rose for a party that includes 50 guests? Enter the answer in the box. Autohide sh Growth: Math 6+ GA 2015 Question ITU Shelf nositi
Answer:
a. Angel has a greater flat fee
b. Difference = $130
Step-by-step explanation:
Given
Rose:
[tex]Flat\ Fee = \$50[/tex]
[tex]Rate = \$12[/tex]
Angel
Guests --- Charges
10 -------- $180
15 -------- $255
25 -------- $405
30 -------- $480
Solving (a): Who charges the greater flat fee?
We have that the flat fee of Rose is:
[tex]Flat\ Fee = \$50[/tex]
For Angel, we need to determine the equation of the given table
Start by calculating the slope (m) of the table
[tex]m = \frac{y_2 - y_1}{x_2 - x_1}[/tex]
Where x and y represent any two corresponding values of the guests and the charges.
[tex](x_1,y_1) = (10, 180)[/tex]
[tex](x_2,y_2) = (30, 480)[/tex]
So: [tex]m = \frac{y_2 - y_1}{x_2 - x_1}[/tex] becomes
[tex]m = \frac{480 - 180}{30 - 10}[/tex]
[tex]m = \frac{300}{20}[/tex]
[tex]m = 15[/tex]
Next, we calculate the equation using:
[tex]y - y_1 = m(x - x_1)[/tex]
Where
[tex]m = 15[/tex]
[tex](x_1,y_1) = (10, 180)[/tex]
[tex]y - 180 = 15(x - 10)[/tex]
[tex]y - 180 = 15x - 150[/tex]
Add 180 to both sides
[tex]y - 180 + 180 = 15x - 150 + 180[/tex]
[tex]y= 15x + 30[/tex]
From the equation above:
The slope = 15 --- This represents the hourly rate
and
y intercept = 30 --- This represents the flat fee
This is better represented as:
Angel
[tex]Flat\ Fee = \$30[/tex]
[tex]Rate = \$15[/tex]
and
Rose:
[tex]Flat\ Fee = \$50[/tex]
[tex]Rate = \$12[/tex]
By comparison, Angel has a greater flat fee
Solving (b): Difference between total charges of 50 guests for both caterers.
For angel, the equation is:
[tex]y= 15x + 30[/tex]
and x = 50.
So:
[tex]y = 15 * 50 + 30[/tex]
[tex]y = 750 + 30[/tex]
[tex]y = 780[/tex]
For Rose,
First, we need to determine the equation.
[tex]Flat\ Fee = \$50[/tex]
[tex]Rate = \$12[/tex]
The equation is:
[tex]y = Flat\ Fee + Rate * x[/tex]
[tex]y = 50 + 12 * x[/tex]
So, the total charges for 50 guests is:
[tex]y = 50 + 12 * 50[/tex]
[tex]y = 50 + 600[/tex]
[tex]y = 650[/tex]
The difference is then calculates as:
[tex]Difference = 780 - 650[/tex]
[tex]Difference = \$130[/tex]
solve the equations 7(x+4)=21
iready
Step-by-step explanation:
[tex]\bullet\hookrightarrow\sf 7 (x+4)=21 [/tex]
[tex]\bullet\Rrightarrow\sf 7x+28=21 [/tex]
[tex]\bullet\rightarrowtail\sf 7x=21-28 [/tex]
[tex]\bullet\twoheadrightarrow\sf 7x=-7 [/tex]
[tex]\bullet\looparrowright\sf x=\dfrac{-7}{7}[/tex]
[tex]\bullet\rightharpoonup\sf x=-1 [/tex]
What is the slope of a line that passes through the following points (4, 6) and (2, 0)?
Answer:
3
Step-by-step explanation:
Step-by-step explanation:
formula to fine slope
m=tan()=y2-y1/x2-x1
Buster needs to find x and y in the following system.
Equation A: - 8x + 14y = 10
Equation B: 4x + 3y = 25
Buster decided to use elimination to solve this system. Do you agree or disagree? Why
Answer:
U GO TO DHS
Step-by-step explanation:
What is the solution of the following system y=-2x=8 16+4x=2y
Question 3 of 5
Which fraction and decimal forms match the long division problem?
SO
tele
OAS 0.50
OR S and 0.58
OGS and 0.58
OS and 0.58
I NEES HELP plz
Answer:
8/15= 0.533
Step-by-step explanation:
8 divided by 15 isn't possible so according to the rules of division we introduce a (0.) then we add a (0) to the back of (8) to make it (80) which is bigger than (15) so we now have (80/15) which is (5) remaining (5) then we add the back of the (0.) it then becomes (0.5) we then step down the remaining (5) and also add a (0) at the back to make it (50) which is also bigger than (15) (50/15) which gives us (3) and a remaining (5) add (3) to the back of (0.5) to become (0.53) also still step down the remaining (5) add a zero to it to become (50) still divide by (15) gives (3) and remains (5) and it gives the same answer so it is a reoccurring decimal.
Thank you hope I was of help!!
Which of the following numbers is closest to 11?
O A. 7123
B. V115
O c. 118
O D. 120
Answer:
the answer is D
Step-by-step explanation:
A. √123 = 11.09
11.09-11 = 0.09
B. √115 = 10.72
11-10.72 = 0.28
C. √118 = 10.86
11-10.86 = 0.14
D. √120 = 10.95
11-10.95 = 0.05 ( the smallest difference )
You need to purchase supplies for an employee appreciation picnic. Your boss gives you the shopping list below and asks that you get the best deal by buying from one grocer, either Quality Groceries or Fresh Market. You call both grocers, and they provide the price information that you put into the table. Which grocer will provide all these picnic supplies for the lowest cost, and what is that cost without sales tax?
Hamburger Patties 50
Hot Dogs 2 dozen
Hamburger Buns 50
Hot Dog Buns 2 dozen
Potato Salad 2 gallons
Ice Cream 5 gallons
Product Quality Groceries Unit Quality Groceries Price per Unit Fresh Market Unit Fresh Market Price per Unit
Hamburger Patties Package of 10 $15.50 Package of 8 $12.40
Hot Dogs Package of 12 $6.23 Package of 10 $4.99
Hamburger Buns Package of 8 $3.79 Package of 8 $3.55
Hot Dog Buns Package of 12 $2.75 Package of 8 $2.55
Potato Salad Gallon $4.33 Gallon $5.55
Ice Cream Gallon $7.55 Gallon $6.99
Answer:
this doesnt make sense is there any type of picture or more explanation on the question? i'll still help :)
Step-by-step explanation:
Find the 5th term of the following sequence:
f(n) = 3(-2)"
Answer:
42
Step-by-step explanation:
uih
How can this expression be written another way?
Apply the distributive property and the greatest common factor to write an equivalent expression.
Enter your answer by filling in the boxes.
105 + 35m = __ (__)
Answer:
(105 x 30) + (105 x 5) + m
Step-by-step explanation:
Answer:
(105 x 30) + (105 x 5) + m
Step-by-step explanation:
Hope this helped have an amazing day!
Use the Distributive Property to expand the expression -3(6.3x + 1.5y)
Answer:
(-3*6.3x)+(-3*1.5y)
(-18.9x)+(-4.5y)
x2 + kx + (k + 3) is tangent to x-axis. Find the value
of k.
Step-by-step explanation:
Since x² + kx + (k + 3) is tangent to the x-axis, the quadratic function has a repeated root.
The discriminant is zero, so b² - 4ac = 0.
=> k² - 4(1)(k + 3) = 0
=> k² - 4k - 12 = 0
=> (k - 6)(k + 2) = 0
Hence k = 6 or k = -2.
The value of k is -2 and 6.
What is an equation?An equation is a mathematical statement that is made up of two expressions connected by an equal sign.
Example:
2x + 4 = 9 is an equation.
We have,
x² + kx + (k + 3) is tangent to the x-axis.
This means,
The equation has roots.
Now,
The determinant of the equation is equal to zero.
D = b² - 4ac
x² + kx + (k + 3) is in the form of ax² + bx + c.
a = 1
b = k
c = k + 3
Now,
D = 0
k² - 4(k + 3) = 0
k² - 4k - 12 = 0
k² - (6 - 2)k - 12 = 0
k² - 6k + 2k - 12 = 0
k(k - 6) + 2(k - 6) = 0
(k + 2) (k - 6) = 0
k + 2 = 0
k = -2
k - 6 = 0
k = 6
Thus,
k = -2 and 6.
Learn more about equations here:
https://brainly.com/question/17194269
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Evaluate 3 + (a + 4)(8 - b) when a = 5 and b = 6.
The value of the expression is
Answer:
21
Step-by-step explanation:
you plug in 5 for a and 6 for b and solve.
The value of the expression 3 + (a + 4)(8 - b) when a = 5 and b = 6 is 21
How to evaluate an expression?3 + (a + 4)(8 - b)
when,
a = 5 and b = 6
substitute into the expression= 3 + (5 + 4) (8 - 6)
= 3 + 9 (2)
= 3 + 18
= 21
Therefore, the value of the expression 3 + (a + 4)(8 - b) when a = 5 and b = 6 is 21
Learn more about evaluation:
https://brainly.com/question/4344214
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Write seven hundred twenty-one in standard form
Answer:
721
Step-by-step explanation:
dog you in elementary school
Need help asap in a hurry, please. Doing a budget The gross income is $54,000.
452356273567187+646735264273852927×648463768÷636478476804387098-3×768370678679868687675785+34579256289574678556+87673976-7568975268565786578467858485647694587897436754675686587897558785578576×830463809745
Answer:
-6.28576e+81
Step-by-step explanation:
please answer 44 mins remaining
Answer:
y=x-2
Hope this helps.
HELPPPPPPPP PLZZZZZZX
Answer:
ur answer should be BStep-by-step explanation:
The mass of a textbook is 1.25 kg about how many pounds is this
Answer:
2.755778
Step-by-step explanation:
Just search how much pounds is 1.25 kg?
ASAP!!!!
What is 4 1/2 x 2
Answer:
Answer is 9 Broski mark as brainliest
Step-by-step explanation:
Answer:
C
Step-by-step explanation:
6.2x1015 (the 15 is above the 10)
Answer:
9.3
Step-by-step explanation:
6.2 x 15/10= 9.3
Answer:
Scientific notation is 1.6 x 10^16 and in regular format is:
160000000000000000