The volume is given by the integral,
[tex]\displaystyle\int_0^{2\pi}\int_0^{\cos^{-1}((\sqrt{65}-1)/8)}\int_0^4\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\varphi\,\mathrm d\theta[/tex]
That is, [tex]\rho[/tex] ranges from the origin to the sphere of radius 4. The range for [tex]\varphi[/tex] starts at the intersection of the cone [tex]z=x^2+y^2[/tex] with the sphere [tex]x^2+y^2+z^2=16[/tex], which gives
[tex]z+z^2=16\implies z^2+z-16=0\implies z=\dfrac{\sqrt{65}-1}2[/tex]
and
[tex]z=4\cos\varphi\implies\varphi=\cos^{-1}\left(\dfrac{\sqrt{65}-1}8\right)[/tex]
and extends to the x-y plane where [tex]\varphi=\frac\pi2[/tex]. The range for [tex]\theta[/tex] is self-evident.
The volume is then
[tex]V=\displaystyle\int_0^{2\pi}\int_0^{\cos^{-1}((\sqrt{65}-1)/8)}\int_0^4\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\varphi\,\mathrm d\theta[/tex]
[tex]V=\displaystyle\left(\int_0^{2\pi}\mathrm d\theta\right)\left(\int_0^{\cos^{-1}((\sqrt{65}-1)/8)}\sin\varphi\,\mathrm d\varphi\right)\left(\int_0^4\rho^2\,\mathrm d\rho\right)[/tex]
[tex]V=2\pi\left(\dfrac{\sqrt{65}-1}8\right)\left(\dfrac{64}3\right)=\boxed{\dfrac{16\pi(9-\sqrt{65})}3}[/tex]
A sphere is a three-dimensional object with a round form. The volume of the sphere is [16π(9-√65)]/3 unit³.
What is a sphere?A sphere is a three-dimensional object with a round form. A sphere, unlike other three-dimensional shapes, has no vertices or edges. Its centre is equidistant from all places on its surface. In other words, the distance between the sphere's centre and any point on its surface is the same.
We know that the volume of the given sphere can be given by the integral,
[tex]{\rm Volume} = \int^{2\pi}_0\int^{cos^{-1}(\frac{\sqrt{65}-1}{8})} \int_0^4\rho^2sin\varphi\ d\rho\ d\varphi\ d\theta[/tex]
where ρ ranges from the origin of the plot to the sphere of radius 4 while the range of φ starts at the intersection of the cone z=x²+y² with the sphere x²+y²+z²=16.
Now, the value of z and φ can be written as,
[tex]x^2+y^2+z^2 = 16\\\\(x^2+y^2)+z^2 = 16\\\\z+z^2 = 16\\\\z^2+z-16=0 \implies z=\dfrac{\sqrt{65}-1}{2}[/tex]
And
[tex]z =4\ cos\ \varphi \implies \varphi =cos^{-1}(\dfrac{\sqrt{65}-1}{8})[/tex]
Further, the volume of the sphere can be written as,
[tex]{\rm Volume} = \int^{2\pi}_0\int^{cos^{-1}(\frac{\sqrt{65}-1}{8})} \int_0^4\rho^2sin\varphi\ d\rho\ d\varphi\ d\theta\\\\\\{\rm Volume} = (\int^{2\pi}_0\ d\theta)(\int^{cos^{-1}(\frac{\sqrt{65}-1}{8})} sin\varphi\ d\varphi)(\int_0^4\rho^2 d\rho)\\\\\\V = 2\pi(\dfrac{\sqrt{65}-1}{8})(\dfrac{64}{3}) = \dfrac{16\pi(9-\sqrt{65})}{3}[/tex]
Hence, the volume of the sphere is [tex]\dfrac{16\pi(9-\sqrt{65})}{3}[/tex].
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Find the slope of the line that passes through the points (3,6) and (5, 3).
-3/2
3/2
2/3
Answer:
-3/2.
Step-by-step explanation:
To find the slope, we find the rise over run.
In this case, the rise is 6 - 3 = 3.
The run is 3 - 5 = -2.
The slope is 3 / (-2) = -3/2.
Hope this helps!
Answer:
-3 / 2
Step-by-step explanation:
slope formula is: change in y / change in x , y2-y1 / x2-x1
let (3.6) is (x1 , y2) and (5 , 3) (x2 , y2)
3 - 6 / 5 - 3 = -3 / 2
the slope is -3 / 2
.. ..
By rounding to one significant figure, estimate the answers to the question circled
A population is increasing according to the exponential function defined by y = 3e^.04x, where y is in millions and x is the number of years. Which of the following should be done in order to answer the question "When will the population reach 4 million?"
A. Solve 3e^.04x=12
B. Evaluate y = 3e^.04x(4).
C. Evaluate y = 3e^.04x(1/4)
D. Solve 3e^.04x= 4.
Answer:
D. Solve 3e^0.04x= 4.Step-by-step explanation:
Given the population increasing according to the exponential function defined by y = 3e^0.04x, where y is in millions and x is the number of years, to determine when the population will reach 4 million, the following steps must be carried out.
Since y is in millions, we will first substitute y = 4 into the modeled equation as shown;
4 = 3e^0.04x
Therefore to calculate x, we need to solve the expression 3e^0.04x = 4 for the value of x and this is arrived at by simply substituting y = 4 into the given equation.
To get x;
3e^0.04x = 4
e^0.04x = 4/3
taking the ln of both sides
lne^0.04x = ln1.333
0.04x = 0.287
x = 0.287/0.04
x≈ 3.59 years
Question 8(Multiple Choice Worth 1 points) (07.01 MC) Find the measure of arc DF. Circle A with chords EF and CD that intersect at point G, the measure of arc EC is 5x plus 10 degrees, the measure of angle EGC is 70 degrees, and the measure of arc DF is 11x plus 2 degrees. 50° 90° 100° 140°
Answer: 90°
Step-by-step explanation:
As known ∡EGC=(arcEC+arcDF)/2
arcEC+arcDF=70°*2
5x+10+11x+2=140
16x+12=140
16x=128
x=128:16
x=8
So arcDF=11*x+2=11*8+2=90°
The measure of arc DF is 90°.
What is circle?A circle is a closed, two-dimensional object where every point in the plane is equally spaced from a central point. The line of reflection symmetry is formed by all lines that traverse the circle. Additionally, every angle has rotational symmetry around the centre.
Given, Circle A with chords EF and CD that intersect at point G,
arc EC = 5x + 10°
arc DF = 11x + 2°
∠EGC = 70°
The measure of the inner angle is the semi-sum of the arcs comprising it and its opposite
so
∠EGC=(1/2)[arc EC+arc DF]
substitute the values
70 = 1/2(5x + 10 + 11x + 2)
70 = 1/2(16x + 12)
140 = 16x + 12
16x = 128
x = 128/16 = 8
so ac DF = 11x + 2
arc DF = 11(8) + 2
arc DF= 88 + 2 = 90°
Hence the value of arc DF is 90°.
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If the sum of the first n terms of an AP is 1/2 (3n^2+7n),then find its nth term.Hence write its 20th term.
Answer:
62
Step-by-step explanation:
Sn=1/2(3n^2+7n)
S1=1/2(3+7)=5
S2=1/2(3*4+7*2)=26/2=13
We know
S1=a1=5
S2=a1+a2=13
S2-s1=a1+a2-a1
13-5=a2
a2=8
We know d=a2-a1
d=8-5=3
nth term of AP =an=5+(n-1)3
an= 2+3n
Therefore 20th term =
a20= 2+3(20)=62
Hence 20th term of AP is 62
I hope this helps
An open box is to be made from a 5 ft by 9 ft rectangular piece of sheet metal by cutting out squares of equal size from the four corners and bending up the sides. Find the maximum volume that the box can have.
Answer: 21 ft³
Step-by-step explanation:
Let x represent the height of the box.
Then the 5 ft width of cardboard is (5 - 2x) when creating the box
and the 9 ft length of cardboard is (9 - 2x) when creating the box.
Volume = length x width x height
= (9 - 2x)(5 - 2x)(x)
= 45x - 28x² + 4x³
Using Calculus to solve for x, set the derivative equal to zero and use the quadratic formula to solve for x:
V' = 45 - 56x + 12x²
0 = 12x² - 56x + 45
x = 3.6, x = 1.0
Use those values to find the width, length, and volume:
height(x) × width (5 - 2x) × length (9 - 2x) = Volume
3.63 × -2.26*
1 × 3 × 7 = 21
*width cannot be negative so the height cannot be 3.63
Please answer this correctly without making mistakes
Answer:
ace hardware store
Step-by-step explanation:
Ace is the place with the helpful hardware folks!
Solve application problems using radical equations. Vince wants to make a square patio in his yard. He has enough concrete to pave an area of 378 square feet. How long can a side of his patio be?
Answer:
3(sqrt42) ft
Step-by-step explanation:
If l is side length of square patio, then area of patio would be l^2.
l^2 = 378
l = sqrt378 = 3(sqrt42)
What is the sum of the reciprocal of 1/-13 and the reciprocal of -1/2
Answer:
-15
Step-by-step explanation:
1/-13
The reciprocal is
[tex]1/ 1/-13[/tex]
Because of the law of indices,
We'll have [tex] 1 * -13 [/tex]
= -13.
-1/2
The reciprocal is
1 ÷ -1/2
Because of the law of indices,
We'll have [tex] 1 * -2 [/tex]
= -2
Therefore, the sum would be -13 + (-2) which would be equals to -15.
-15
Step-by-step explanation:
1/-13
The reciprocal is
Because of the law of indices,
We'll have
= -13.
-1/2
The reciprocal is
1 ÷ -1/2
Because of the law of indices,
We'll have
= -2
Therefore, the sum would be -13 + (-2) which would be equals to -15.
ASAP 300* POINTS PLUS BRAINLIEST!!! ONLY IF THE ANSWER IS CORRECT!!! What is the value of this expression when a = 3 and b = negative 2? (StartFraction 3 a Superscript negative 2 Baseline b Superscript 6 Baseline Over 2 a Superscript negative 1 Baseline b Superscript 5 Baseline EndFraction) squared? THERE ARE NO OPTIONS FOR THIS QUESTION MUST SHOW STEPS!!!
*/20, or 300/20.
Answer:
The value of this expression when a = 3 and b = -2 is -1.
Step-by-step explanation:
The expression is:
[tex]X=\frac{3a^{-2}b^{6}}{2a^{-1}b^{5}}[/tex]
Compute the value of X for a = 3 and b = -2 as follows:
[tex]X=\frac{3a^{-2}b^{6}}{2a^{-1}b^{5}}[/tex]
[tex]=\frac{3\cdot (3)^{-2}\cdot (-2)^{6}}{2\cdot (3)^{-1}\cdot (-2)^{5}}\\\\=\frac{(3^{-1})\times (-2)^{6}}{2\cdot (3^{-1})\cdot (-2)^{5}}\\\\=\frac{(-2)^{1}}{2}\\\\=-1[/tex]
Thus, the value of this expression when a = 3 and b = -2 is -1.
1). f(x) = 3x + 15 then what's f^-1(x)?
Answer:
Step-by-step explanation:
f(x)=3x+15
let f(x)=y
y=3x+15
flip x and y
x=3y+15
3y=x-15
y=1/3 x-5
or f^{-1}x=1/3 x-5
what is the solution to the system of equations?
y = 4x-10
y=2
A) (3,2)
B) (2,3)
C) (-2,2)
D) (2,-2)
Answer:
A) (3, 2)
Step-by-step explanation:
y = 4x - 10
y = 2
Substitute y with 2 in the first equation and solve for x.
y = 4x - 10
2 = 4x - 10
12 = 4x
3 = x
Solution: x = 3, y = 2
Answer: A) (3, 2)
A storage basin is 1240 ft by 637 ft. It’s 15.63 ft deep. How many gallons of water is in the basin
Answer:
12345824.4
Step-by-step explanation:
I'm not really sure. I just did 1240x637x15.63=12345824.4
answer:
53,445.1 gallons of water
Step-by-step explanation:
im not exactly sure but i looked up how to calculate gallons in a container and the website i went to told me to measure the interior length, width, and height which was already done, then it told me to multiply the length by width by height to get the volume of the container then it told me to divide the volume by 231 to get the number of gallons in the container, so 1,240* 637= 789,880* 15.63= 12,345,824.4÷231= 53,445.127272727272727272727272727 which rounded to the nearest tenth is 53,445.1, i honestly hope that this is right and that it helps
Find the slope of the line that passes through (1, 14) and (4,9)
Which two numbers in the points represent x values? Select both in the
list.
In any coordinate pair, the first number is the x-value and the second number is the y-value.
To find the slope, simply take the difference of the y values and divide by the difference in the x values: (14-9)/(1-4) is equal to -5/3.
The slope of the line that passes through (1, 14) and (4,9) is -5/3.
It is find the slope of the line.
what is slope?The slope of any line, ray, or line segment is the ratio of the vertical to the horizontal distance between any two points on it (“slope equals rise over run”).
The slope is always calculated from the rise divided by the run. Typically, the equation is presented as:
m = Rise/Run
If you have two points, the points should be [tex]P_{1} (x_{1} ,y_{1} )[/tex] and [tex]P_{2} (x_{2} ,y_{2} )[/tex] So, the equation would be:
[tex]m=\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]
In any coordinate pair, the first number is the x-value and the second number is the y-value.
The difference of the y values and divide by the difference in the x values:
m=(14-9)/(1-4) is equal to -5/3.
The slope of the line that passes through (1, 14) and (4,9) is -5/3.
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Please answer this correctly without making mistakes
Please simplify the correct answer
Hi there!! (✿◕‿◕)
⭐⭐⭐⭐⭐⭐⭐⭐⭐⭐⭐⭐⭐⭐⭐⭐⭐⭐⭐⭐⭐⭐
Hillsboro: 700 houses sold
Lowell: 100 houses sold
700 + 100 = 800
Hillsboro and Lowell: 800 houses sold
Other: 600
800 + 600 = 1,400
Clay County: 1,400 houses sold
800/1,400 = 4/7
Hope this helped!! ٩(◕‿◕｡)۶
Answer:
4/7
Step-by-step explanation:
Well first we need to find the total amount of houses sold in Clay County.
700 + 100 + 600
= 1,400
Now we need to find the total of houses sold in Lowell and Hillsboro.
700 + 100
= 800
Now we can make the fraction,
800/1400
and simplify
400/700
200/350
20/35
4/7
Thus,
the fraction of houses sold in Hillsboro and Lowell in Clay County is 4/7.
Hope this helps :)
In June, Jasmine earned $1200 in commission. In July, her commission decreased by 7% Write an expression that can be used to find her commission in July.
Answer:
$1116
Step-by-step explanation:
An infinite population has a standard deviation of 10. A random sample of 100 items from this population is selected. The sample mean is determined to be 60. At 80% confidence, the margin of error is
Answer:
1.28
Step-by-step explanation:
Margin of error is also called confidence interval, and it shows how much the results from sample reflects the characteristics of the larger population.
The following steps are used to calculate margin of error.
- obtain population standard deviation (σ) and sample size (n)
- divide standard deviation by square root of sample size
- multiply result by z score consistent with the given confidence. In this case confidence is 80% and z score for it is 1.28
Margin of error= z * (σ ÷ √n)
Margin of error = 1.28 * (10 ÷ √100)
Margin of error= 1.28 * (10 ÷ 10) = 1.28
Winston and Alice are taking a trip. Winston left at 8 am and traveled an average of 50 miles per hour. Alice left at 10 am and traveled an average of 70 miles per hour. At what time are they at the same place at the same time? Write a system of equation to represent this situation. Then use the substitution method with that system to determine at the time they will be in the same location. How many miles away from home will they be at that time?
Answer:
3 PM
350 miles
Step-by-step explanation:
Let's say t is the number of hours since 8 AM.
The distance traveled by Winston is:
w = 50t
The distance traveled by Alice is:
a = 70(t−2)
When w = a:
50t = 70(t−2)
50t = 70t − 140
140 = 20t
t = 7
Winston and Alice will be at the same place 7 hours after 8 AM, or 3 PM.
The distance they travel is 350 miles.
The line passing through points
(4,0) and (-2, 1) has a slope of?
A. -6
B. -1/6
C. 1/2
D. 2
E. 1/6
Answer:
b. -1/6
Step-by-step explanation:
slope = (difference in y)/(difference in x)
slope = (1 - 0)/(-2 - 4) = 1/(-6) = -1/6
Answer:
m = -1/6 = B
Step-by-step explanation:
[tex]m = \frac{y_2-y_1}{x_2-x_1} \\ x_1=4\\ y_1=0\\ x_2=-2\\y_2=1.\\m = \frac{1-0}{-2-4} \\m = \frac{1}{-6}[/tex]
Rent for a 3 bedroom apartment is regularly $936 per month. Apartment management is now offering one month free for a 12 month lease. If you sign a one year lease and apply the free month equally across months, how much is your monthly lease amount?
Answer:
$858
Step-by-step explanation:
Monthly lease amount is ⇒ $936
Considering one moth free, yearly rental would be
⇒ 11*$936= $10296The monthly lease amount :
⇒ $10296 / 12 = $858Determine whether each red path in the vector field is positive, negative, or zero
If the path is perpendicular to the field it is zero.
If the path is along the field it is positive or negative depending on it's direction.
See the attached picture.
Researchers at the Centers for Disease Control and Prevention have been studying the decay pattern of a new virus with a decay rate of 22% per hour. They start with 500 viruses that they want to check on in the next 8 hours. How many viruses will they find in 8 hours? Round your answer to the nearest whole number.
Answer:
They will find 69 viruses in 8 hours.
Step-by-step explanation:
The number of viruses after t hours is given by the following equation:
[tex]V(t) = V(0)(1-r)^{t}[/tex]
In which V(0) is the initial number of viruses and r is the decay rate, as a decimal.
They start with 500 viruses
This means that [tex]V(0) = 500[/tex]
Decay rate of 22% per hour.
This means that [tex]r = 0.22[/tex]
So
[tex]V(t) = V(0)(1-r)^{t}[/tex]
[tex]V(t) = 500(1-0.22)^{t}[/tex]
[tex]V(t) = 500(0.78)^{t}[/tex]
How many viruses will they find in 8 hours?
This is V(8).
[tex]V(t) = 500(0.78)^{t}[/tex]
[tex]V(8) = 500(0.78)^{8}[/tex]
[tex]V(8) = 68.51[/tex]
Rounding to the nearest whole number
They will find 69 viruses in 8 hours.
Answer:
The researchers will find 86 viruses.
Step-by-step explanation:
Identify the value of each variable in the formula. Be sure to put the percent in decimal form. Be sure the units match—the rate is per hour and the time is in hours.
A =?
A0 = 500
R= -0.22/hour
t= 8 hours
Substitute the values in the formula.
A= A0e^rt
A= 500e^-0.22x8
Compute the amount.
A ≈ 86.02
Round to the nearest whole number.
A ≈ 86
Find the volume of the cylinder.
The approximate volume only applies when pi = 3.14
Use either answer, but not both of course.
===============================================
Work Shown:
V = volume of cylinder
V = pi*r^2*h
V = pi*2^2*8
V = pi*32
V = 32pi .... exact volume in terms of pi
V = 32*3.14
V = 100.48 .... approximate volume when we use pi = 3.14
A manufacturing process that produces electron tubes is known to have a 10% defective rate. Suppose a random sample of 15 tubes is selected from the manufacturing process. a) Find the probability that no more than two defectives are found?
Answer:
Probability of obtaining no more than two defective tubes = 0.816
Step-by-step explanation:
The Probability of obtaining no more than two defective tubes in a randomly selected sample of 15 tubes is obtained using the binomial distribution formula: nCr × p^r × q^(n -r).
Where n is number of samples;
r is maximum number of defective tubes, r ≤ 2;
p is probability of defective tubes = 10% or 0.1
q is probability of non-defective tubes, q = 1 - p
Further explanations and calculations are given in the attachment below:
A pharmacist wants to mix a 30% saline solution with a 10% saline solution to get 200 mL of a 12% saline solution. How much of each solution should she use
Answer:
30% constituents=20 mL
10% constituents=180 mL
Step-by-step explanation:
x= 30% volume
y=10% volume
For our first equation, we know the total volume is 200 mL and is the sum
x+y=200
y=200-x (1)
For our second equation, we do a mass balance for 200 mL of final solution.
12% w/v = 0.12 g/mL
This means that in 1 mL of solution, we have 0.12 g of NaCl.
For any solution, concentration multiplied by volume will give the mass of NaCl:
Mass in x mL= C*V (g/mL) (mL)
So in 200 mL, we have
0.12*200 (g/mL) (mL)
=24g of NaCl
Cx*Vx + Cy*Vy=24
0.3x+0.1y=24 (2)
Substitute y=200-x into (2)
0.3x+0.1(200-x)=24
0.3x+20-0.1x=24
0.2x=24-20
0.2x=4
Divide both sides by 0.2
0.2x/0.2=4/0.2
x=20
Substitute x=20 into (1)
y=200-x
y=200-20
y=180
30% constituents=20 mL
10% constituents=180 mL
The square below represents one whole.
Answer:
6 percent
Step-by-step explanation:
6/100 is shaded. This is the same thing as 0.06. Once I move the decimal two places to the right, I get 6. (I move the decimal place because percents are out of 100, so two decimal places.) This means 6 percent.
Find the perimeter and total area of the polygon shape shown below. All measurements are given in inches. Helps please !!!!
Answer:
perimeter = 56 in
area = 192 sq. in.
Step-by-step explanation:
area of a triangle = 0.5 * b * h
b = 12
h = 8
At = (0.5 * 12 * 8) = 48
Area of a square
As = 12 * 12 = 144
total area = At + As
total area = 48 + 144
total area = 192 sq. in.
perimeter = add all sides
12 + 12 + 12 + 10 + 10 = 56 in
hope it helps
2.35=11x Equals What
Answer:
x=0.2136
Step-by-step explanation:
Answer:
x=0.214 rounded to the thousandths
Step-by-step explanation:
2.35=11x
divide each side by 11 to isolate the x
x=0.214 rounded to the thousandths
Can you draw the reflection Across the y-axis of the attached image.
Answer:
see graph
Step-by-step explanation:
A reflection across the y-axis means the point is equal but opposite distance from the y-axis. This has no change on the y-value of the point, because no matter the y-value, the point will still be the same distance from the y-axis. Long story short, if you're reflecting across the y-axis, change the sign of the x-coordinate. If you're reflecting across the x- axis, change the sign of the y-coordinate.
if 176 dollars can be exchanged for the 140pounds and 1 pound is exchanged for RS 130 how many dollars can be exchanged for RS 2925
Answer:
Number of dollar = 28.28 (Approx)
Step-by-step explanation:
Given:
176 dollar = 140 pound
1 pound = Rs.130
Find:
Number of dollars in Rs 2925
Computation:
1 pound = Rs.130
Number of pound = 2925 / 130
Number of pound = 22.5 pound
1 pound = 176 / 140
Number of dollar = [176 / 140]22.5
Number of dollar = 28.28 (Approx)