Use spherical coordinates. let h be a solid hemisphere of radius 1 whose density at any point is proportional to its distance from the center of the base. (let k be the constant of proportionality.) (a) find the mass of h.
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The full answer in attachment, we use the partial integral using the spherical coordinates,
and we find that the mass of h = πK/2
To find the mass of the solid hemisphere using spherical coordinates with density proportional to distance from center, integrate the density times the volume element over the hemisphere.
Explanation:To find the mass of the solid hemisphere using spherical coordinates, we first need to understand that the density at any point is proportional to its distance from the center of the base. Let's denote the constant of proportionality as k. The mass can be found by integrating the product of density and volume over the entire hemisphere. The volume element in spherical coordinates is given by ρ² sinϕ dρ dϕ dθ, where ρ is the radial distance, ϕ is the polar angle, and θ is the azimuthal angle.
We can denote the density as ρ = kρ, where ρ is the radial distance from the center. The mass element dm is then given by dm = ρ² sinϕ dρ dϕ dθ = kρ³ sinϕ dρ dϕ dθ. To obtain the mass of the hemisphere, we need to integrate the mass element over the appropriate limits, which are ρ = 0 to 1, ϕ = 0 to π/2, and θ = 0 to 2π.
Performing the integration, we get the mass as M = ∫∫∫ kρ³ sinϕ dρ dϕ dθ = πk/6.
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Related Questions
What is 8640minutes in days?(remember ,there are 60minutes in an hour and 24 in a day
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need help to do this
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6. The horizontal and vertical asymptotes are y=1, x=4, and there is a "hole" at x=-2. The appropriate choice is ...
F. Oblique asymptote y = 0
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J. As x → ±∞, f(x) → 0
Hey can you please help me posted picture of question
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First shooting a basketball. Two options: X or O
Then rolling a die. Six options: 1, 2, 3, 4, 5, or 6
Sample space:
X1
X2
X3
X4
X5
X6
O1
O2
O3
O4
O5
O6
Answer: Options:
A. X1
C. O6
F. X6
look at clock blair arrived a the bus stop at 45 minutes ago what time did blaire arive at the bus stop?
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hey can you please help me posted picture of question
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Diagram D
Explanation:
1- let's get the base function:
We are given that:
f(6) = 9
f(7) = 10
f(8) = 11
f(9) = 12
From these givens, we can note that the function can be represented as follows:
f(x) = x + 3
2- let's get the inverse:
To get the inverse of a function, we will simply swap the x and y variable
This can be done as follows:
f(x) = x + 3
y = x + 3
Now, we swap:
x = y + 3
x - 3 = y + 3 - 3
y = x - 3
3- let's now get the values for the given x:
f(6) = 6 - 3 = 3
f(7) = 7 - 3 = 4
f(8) = 8 - 3 = 5
f(9) = 9 - 3 = 6
f(10) = 10 - 3 = 7
f(11) = 11 - 3 = 8
f(12) = 12 - 3 = 9
Comparing the calculated values with the given diagrams, we can note that the correct diagram is D
Hope this helps :)
Trica's mother let her play outside for 15 minutes. When she went outside,she played with her dog for 5 minutes.Then she rode her bike for 4 minutes .She spent the rest of her time catching bugs.how many minutes did Trica spent catching bugs?
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two angles are supplementary. One of the angles has a measure of 60 degrees. What is the measure of the other angle?
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Answer:
120°
Step-by-step explanation:
Consider the following grsph of a quadratic function.
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[tex]y=\frac{7}{8}\left(x+4\right)\left(x+2\right)[/tex]
Find the sum of these polynomials.
(x2 – x + 7) + (9x2 + 6) =
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[tex]( x^{2} -x+7)+(9 x^{2} +6)[/tex]
First work out the parenthesis.
[tex] x^{2} -x+7+9 x^{2} +6[/tex]
Rearrange the expression.
[tex] x^{2} +9 x^{2} -x+7+6[/tex]
Now collect the terms.
[tex]10 x^{2} - x+13[/tex]
The answer is B.
~ Hope this helps you!
Answer:B
Step-by-step explanation:
To date, Ty has cumulative earnings of $107,600. This week he is paid $3,000. What's the total amount of social security tax for this week? Assume a rate of 6.2% on $110,100 for social security and 1.45% for Medicare.
A. $186.00
B. $155.00
C. $180.00
D. $57.66
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Final answer:
The total Social Security tax for Ty's earnings this week, given his total earnings are now $110,600, is $155.00. This calculation considers the Social Security tax rate of 6.2% up to the income threshold of $110,100. Thus, option B is correct.
Explanation:
The question involves calculating the Social Security tax for Ty's earnings this week. With cumulative earnings to date at $107,600 and an additional $3,000 this week, Ty's total earnings for the year are now $110,600.
Since the Social Security tax rate is 6.2% and applies only up to an income of $110,100, we will only apply the tax rate to the $110,100 threshold, not the full $110,600. The week's Social Security tax is then:
0.062 × $110,100 = $6,826.20 (for the entire year so far)
Since we only need to find out the tax for this week's earnings, we need to calculate the tax that was already paid on the first $107,600:
0.062 × $107,600 = $6,671.20
The additional Social Security tax for this week is the difference between the two:
$6,826.20 - $6,671.20 = $155.00
Therefore, the total Social Security tax for this week is $155.00, which corresponds to option B.
The total amount of Social Security tax for Ty this week is $155.00. Hence, the correct option is B.
To calculate the total amount of Social Security tax for Ty this week, we must apply the 6.2% rate to his weekly pay of $3,000. However, since his cumulative earnings to date are $107,600 and the Social Security tax applies only up to $110,100, we need to determine how much of his $3,000 weekly salary is subject to this tax.
To find this, we subtract his current earnings from the maximum amount that is subject to Social Security tax: $110,100 - $107,600 = $2,500. Only $2,500 of Ty's $3,000 salary is taxable for Social Security purposes since he has reached the threshold beyond which Social Security tax is not levied on additional income.
Therefore, this week's Social Security tax is 6.2% of $2,500, which equals $155.00. As the rate for Medicare does not cap, the full $3,000 is subject to the Medicare tax at a rate of 1.45%, leading to a Medicare tax amount of $43.50. However, as the question only asks for the Social Security tax, the answer is B. $155.00.
Find the slope and y-intercept fotv
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Slope: 2/3
The line crosses the y-axis at y=0.
y-intercept: 0
Express the number as a ratio of integers. 3.469 = 3.469469469
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3.469469469....
then:
Let x=3.469469469.....
then
1000x-x=(3469.469469469...) - (3.469469469....)
999x = 3466
x=3466 / 999
= 3+469/999
[tex]=3\frac{469}{999}[/tex]
The function f(t) = 25 sin (pi over 2t) + 10 models the temperature of a periodic chemical reaction where t represents time in hours. What are the maximum and minimum temperatures of the reaction, and how long does the entire cycle take? (5 points)
A. Maximum: 25°; minimum: 10°; period: pi over 2 hours
B. Maximum: 35°; minimum: 35°; period: 8 hours
C. Maximum: 100°; minimum: −15°; period: pi over 2 hours
D. Maximum: 35°; minimum: −15°; period: 4 hours
Answers
By graphing the given function
The correct option is D
D. Maximum: 35°; minimum: −15°; period: 4 hours
Answer:
The correct answer is: Option: D
D. Maximum: 35° ; minimum: −15° ; Period: 4 hours
Step-by-step explanation:
We are given a function f(t) as:
[tex]f(t)=25\sin (\dfrac{\pi}{2}t)+10[/tex]
We know that the maximum value of the function is obtained when the sine function is maximum.
i.e. when sine takes the value=1
and the minimum value of the function is obtained when the sine function is minimum.
i.e. when sine takes the value= -1
Hence, the minimum value of the function is:
-25+10= -15
and the maximum value of the function is:
25+10=35
Also the period of the function of the type:
[tex]f(x)=a\sin(bx)+c[/tex]
is:
[tex]Period=\dfrac{2\pi}{b}[/tex]
Hence, here we have: b=π/2
Hence, period is:
[tex]Period=\dfrac{2\pi}{\dfrac{\pi}{2}}\\\\\\Period=4[/tex]
Hence, option: D is the correct answer.
The solid shown are composed of prisms and half cylinders. Find the volume of the solid. Round to the nearest whole number.
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There are 3 teams of 18 employees working today. Company
policy says that we need to have 1 supervisor for every 8 employees on the floor. We need ....
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Final answer:
The total number of employees working is 54. According to company policy requiring 1 supervisor for every 8 employees, 7 supervisors are needed to comply with this policy.
Explanation:
To find how many supervisors are required, we first need to know the total number of employees working. Since there are 3 teams of 18 employees, the total number of employees is 3 teams * 18 employees per team = 54 employees. Company policy requires 1 supervisor for every 8 employees. Therefore, we divide the total number of employees by the supervisor to employee ratio: 54 employees / 8 employees per supervisor = 6.75. Since we cannot have a fraction of a supervisor, we round up to the next whole number, making it 7 supervisors needed in total.
So, the supervisor would say: "We need 7 supervisors working tonight so we should be OK."
you deposit $3000 into a bank account that pays 1.25% annual interest, compounded semi-annually. how much interest does the account earn after 4 years?
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To find the amount of interest earned on a bank account after 4 years, use the compound interest formula. After plugging in the values, the account earns approximately $162.89 in interest.
Explanation:To calculate the amount of interest earned on a bank account after 4 years, we need to use the formula for compound interest: A = P(1 + r/n)^(nt) - P. In this case, the initial deposit (P) is $3000, the annual interest rate (r) is 1.25%, compounded semi-annually (n = 2), and the time period (t) is 4 years. Plugging in these values, we get:
A = 3000(1 + 0.0125/2)^(2*4) - 3000
Solving this equation, we find that the account earns approximately $162.89 in interest after 4 years.
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SOMEONE PLEASE PLEASE HELP ASAP
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To find a sequence is arithmetic, we are going to find its common difference: [tex]d=a_{n}-a_{n-1}[/tex]
where
[tex]d[/tex] is the common difference
[tex]a_{n}[/tex] is the current term in the sequence
[tex]a_{n-1}[/tex] is the previous term in the sequence
To find if a sequence is geometric, we are going to find its common ratio:
[tex]r= \frac{a_{n}}{a_{n-1}} [/tex]
where
[tex]r[/tex] is the common ratio
[tex]a_{n}[/tex] is the current term in the sequence
[tex]a_{n-1}[/tex] is the previous term in the sequence
1) 3,4,6,10,18
For [tex]a_{n}=4[/tex] and [tex]a_{n-1}=3[/tex]:
[tex]d=a_{n}-a_{n-1}[/tex]
[tex]d=4-3[/tex]
[tex]d=1[/tex]
For [tex]a_{n}=6[/tex] and [tex]a_{n-1}=4[/tex]:
[tex]d=6-4[/tex]
[tex]d=2[/tex]
For [tex]a_{n}=4[/tex] and [tex]a_{n-1}=3[/tex]:
[tex]r= \frac{a_{n}}{a_{n-1}} [/tex]
[tex]r= \frac{4}{3} [/tex]
For [tex]a_{n}=6[/tex] and [tex]a_{n-1}=4[/tex]:
[tex]r= \frac{6}{4} [/tex]
[tex]r= \frac{3}{2} [/tex]
We can conclude that the sequence is neither arithmetic nor geometric.
2) [tex] \frac{25}{4} , \frac{5}{2} ,1...[/tex]
For [tex]a_{n}= \frac{5}{2} [/tex] and [tex]a_{n-1}= \frac{25}{4} [/tex]:
[tex]d=a_{n}-a_{n-1}[/tex]
[tex]d=\frac{5}{2}-\frac{25}{4}[/tex]
[tex]d=- \frac{15}{4} [/tex]
For [tex]a_{n}=1[/tex] and [tex]a_{n-1}= \frac{5}{2} [/tex]:
[tex]d= 1-\frac{5}{2}[/tex]
[tex]d= -\frac{3}{2} [/tex]
For [tex]a_{n}= \frac{5}{2} [/tex] and [tex]a_{n-1}= \frac{25}{4} [/tex]:
[tex]r= \frac{a_{n}}{a_{n-1}} [/tex]
[tex]r= \frac{\frac{5}{2} }{\frac{25}{4} } [/tex]
[tex]r= \frac{2}{5} [/tex]
For [tex]a_{n}=1[/tex] and [tex]a_{n-1}= \frac{5}{2} [/tex]:
[tex]r= \frac{1}{\frac{5}{2}} [/tex]
[tex]r= \frac{2}{5} [/tex]
We have a common ratio, we can conclude that the sequence is geometric.
3) [tex] \frac{2}{3} , \frac{4}{6} , \frac{8}{9} ...[/tex]
For [tex]a_{n}=\frac{4}{6}[/tex] and [tex]a_{n-1}= \frac{2}{3}[/tex]:
[tex]d=\frac{4}{6}-\frac{2}{3}[/tex]
[tex]d=0[/tex]
For [tex]a_{n}=\frac{4}{6}[/tex] and [tex]a_{n-1}= \frac{2}{3}[/tex]:
[tex]r= \frac{\frac{4}{6}}{\frac{2}{3}} [/tex]
[tex]r=1[/tex]
For [tex]a_{n}=\frac{8}{9}[/tex] and [tex]a_{n-1}= \frac{4}{6}[/tex]:
[tex]r= \frac{\frac{8}{9}}{\frac{4}{6}} [/tex]
[tex]r= \frac{4}{3} [/tex]
We can conclude that the sequence is neither arithmetic nor geometric.
4) 3,15,75...
For [tex]a_{n}=15[/tex] and [tex]a_{n-1}=3[/tex]:
[tex]d=15-3[/tex]
[tex]d=12[/tex]
For [tex]a_{n}=75[/tex] and [tex]a_{n-1}=15[/tex]:
[tex]d=75-15[/tex]
[tex]d=60[/tex]
For [tex]a_{n}=15[/tex] and [tex]a_{n-1}=3[/tex]:
[tex]r= \frac{15}{3} [/tex]
[tex]r=5[/tex]
For [tex]a_{n}=75[/tex] and [tex]a_{n-1}=15[/tex]:
[tex]r= \frac{75}{15} [/tex]
[tex]r=5[/tex]
We have a common ratio, we can conclude that the sequence is geometric.
5) 5,-11,-27
For [tex]a_{n}=-11[/tex] and [tex]a_{n-1}=5[/tex]:
[tex]d=-11-5[/tex]
[tex]d=-16[/tex]
For [tex]a_{n}=-27[/tex] and [tex]a_{n-1}=-11[/tex]:
[tex]d=-27--11[/tex]
[tex]d=-27+11[/tex]
[tex]d=-16[/tex]
We have a common difference, we can conclude that the sequence is arithmetic.
The sum of first three terms of a finite geometric series is -7/10 and their product is -1/125. [Hint: Use a/r, a, and ar to represent the first three terms, respectively.] The three number are __, __, and __
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geometric sequences can be represented as
[tex]a_n=a(r)^{n-1}[/tex]
so the first 3 terms are
[tex]a_1=a[/tex]
[tex]a_2=a(r)[/tex]
[tex]a_2=a(r)^2[/tex]
the sum is -7/10
[tex]\frac{-7}{10}=a+ar+ar^2[/tex]
and their product is -1/125
[tex]\frac{-1}{125}=(a)(ar)(ar^2)=a^3r^3=(ar)^3[/tex]
from the 2nd equation we can take the cube root of both sides to get
[tex]\frac{-1}{5}=ar[/tex]
note that a=ar/r and ar²=(ar)r
so now rewrite 1st equation as
[tex]\frac{-7}{10}=\frac{ar}{r}+ar+(ar)r[/tex]
subsituting -1/5 for ar
[tex]\frac{-7}{10}=\frac{\frac{-1}{5}}{r}+\frac{-1}{5}+(\frac{-1}{5})r[/tex]
which simplifies to
[tex]\frac{-7}{10}=\frac{-1}{5r}+\frac{-1}{5}+\frac{-r}{5}[/tex]
multiply both sides by 10r
-7r=-2-2r-2r²
add (2r²+2r+2) to both sides
2r²-5r+2=0
solve using quadratic formula
for [tex]ax^2+bx+c=0[/tex]
[tex]x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}[/tex]
so
for 2r²-5r+2=0
a=2
b=-5
c=2
[tex]r=\frac{-(-5) \pm \sqrt{(-5)^2-4(2)(2)}}{2(2)}[/tex]
[tex]r=\frac{5 \pm \sqrt{25-16}}{4}[/tex]
[tex]r=\frac{5 \pm \sqrt{9}}{4}[/tex]
[tex]r=\frac{5 \pm 3}{4}[/tex]
so
[tex]r=\frac{5+3}{4}=\frac{8}{4}=2[/tex] or [tex]r=\frac{5-3}{4}=\frac{2}{4}=\frac{1}{2}[/tex]
use them to solve for the value of a
[tex]\frac{-1}{5}=ar[/tex]
[tex]\frac{-1}{5r}=a[/tex]
try for r=2 and 1/2
[tex] a=\frac{-1}{10}[/tex] or [tex]a=\frac{-2}{5}[/tex]
test each
for a=-1/10 and r=2
a+ar+ar²=[tex]\frac{-1}{10}+\frac{-2}{10}+\frac{-4}{10}=\frac{-7}{10}[/tex]
it works
for a=-2/5 and r=1/2
a+ar+ar²=[tex]\frac{-2}{5}+\frac{-1}{5}+\frac{-1}{10}=\frac{-7}{10}[/tex]
it works
both have the same terms but one is simplified
the 3 numbers are [tex]\frac{-2}{5}[/tex], [tex]\frac{-1}{5}[/tex], and [tex]\frac{-1}{10}[/tex]
Answer:
2r²-5r+2=0
a=2
b=-5
c=2
Step-by-step explanation:
the volume of a rectangle is 672 inches. the hieght of the prism is 8 incheswhich measurement in inches could be the dimensions of the base?explain and justify your answer
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The volume of a rectangular prism is 672 in³, the height of the prism is 8 inches which measurement in inches could be the dimensions of the base?explain and justify your answer
we know that
volume of a rectangular prism=area of the base*height of the prism
area of the base=Volume/height
Volume=672 in³
height=8 in
area of the base=672/8----> 84 in²
This means that the length and width can be any pair of numbers that has a product of 84 in²
examples
8*8, 9*64/9, 2*32, 16*4, etc
Keep in mind that the dimensions not necessarily are whole numbers, there are actually infinitely answers to this problem
The answer is 6 inches
Hey can you please help me posted picture of question
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____________________
Addition of a number to x indicates a shift towards left. Since 4 is being added to x, this means function is shifted to left by 4 units.
Addition of a number to the function represent an upward shift. Since 2 is being added to the function value, this mean function is shifted up by 2 units.
So, G(x) is obtained by shifted F(x) 4 units to left and 2 units up.
Thus, option C is the correct answer
name the shaded region in three equivalent forms 2/3 in to a fraction,decimal, and percent
Answers
Decimal: .33 (for middle schoolers I suppose)
Percent: 33% (Please Learn you math)
I hope this helped you
A particular model of walkie talkie can broadcast in a circular .the radius of the broadcast area is 10,000 feet. find the area of the circle to the nearest square foot use 3.14 for pi
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A = 3.14 * 10000 * 10000 = 314000000 ft^2
What proportion is more than 2.0 standard deviations from the mean?
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Final answer:
The proportion of data more than 2.0 standard deviations away from the mean in a normal distribution is less than 5%, consisting of 2.5% in the upper tail and 2.5% in the lower tail.
Explanation:
The proportion of values more than 2.0 standard deviations from the mean in a normal distribution is less than 5% because according to the empirical rule, 95% of the data falls within 2 standard deviations of the mean. Therefore, if you are looking at a normal distribution and you want to know the proportion of data that lies beyond 2 standard deviations from the mean, you would subtract 95% from 100%, which is 5%. This is then split between both tails of the distribution, so you have 2.5% in the upper tail and 2.5% in the lower tail, for a total of 5% more than 2 standard deviations away from the mean.
icole wants to buy a pair of jeans. The original cost of the jeans is $42.50, and the markup is 10 percent. How much will she have to pay for the jeans? $44.50 $46.75 $52.50
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42.50 + 4.25 = 46.75.
She will have to pay $46.75 for the jeans.
Sierra's swimming lessons last 1 hour 20 minutes.she finished her lesson at 10:50 am.at what time did her lesson start?
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if the time she finished is 10.50 and her lessons are an hour and 20 minutes long you subtract the finishing time to the time it takes for her lesson.
10.50-1.20=9.30
just used a calculator but mental math is usable for this problem.
LOVE YOU
The time at which her lesson start is 9:30 am.
How are number of people to time needed to complete a task related?More people to do a task means less time it will take.
Less people to do a task means more time it will take.
Thus, they are inversely related.
If x men take y time for a work,
We can define a constant as "Manpower" needed for doing that specific work.
Let we define:
Manpower needed for a work = y + y + y + .. + y = Time per man × count of men
Manpower needed for a work = xy
We are given that;
Time= 1 hour 20 minutes
Now,
To find the time when Sierra’s lesson started, you need to subtract the duration of her lesson from the time when she finished. You can do this by subtracting the minutes and then the hours.
First, subtract 20 minutes from 10:50 am. To do this, you can take away 10 minutes to get 10:40 am and then take away another 10 minutes to get 10:30 am.
Next, subtract 1 hour from 10:30 am. To do this, you can take away 60 minutes or simply reduce the hour by 1. Either way, you get 9:30 am.
Therefore, by work and time the answer will be 9:30 am.
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you and your friend are both knitting scarves for charity . you knit 8 rows each minute and have already knitted 10 rows . your friend knits 5 rows each minute and has already knitted 19 rows . when will you both have knitted the same amount of rows ?
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5, 10, 15, 20, 25, 30, 35, 40 (after 8 minutes)
8, 16, 24, 32, 40 (after 5 minutes)
You and your friend will knit the same number of rows after 8 minutes for you and 5 minutes for your friend.
After 3 minutes you both have knitted the same amount of rows.
What is a numerical expression?A numerical expression is a mathematical statement written in the form of numbers and unknown variables. We can form numerical expressions from statements.
Given, You knit 8 rows each minute and have already knitted 10 rows.
Your friend knits 5 rows each minute and has already knitted 19 rows.
Assuming t is the time when will you both have knitted the same amount of rows,
8t + 10 = 5t + 19.
8t - 5t = 19 - 10.
3t = 9.
t = 3 minutes.
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Expression for calculation divided 10 in half and subtract from 50 divided into fifth
Answers
so 10/2=
5
50/5
10
subtract 5-10
-5 is your answer
^o^
Find the measure of each angle
Answers
9 = 180 - 60 = 120
10 = 180 -80 = 100
6 = 9 = 120
5 = 60
4 = 80
7 = 10 = 100
1=60
3 = 80
2 = 180 - 60 - 80 = 40
8 = 1 = 60
please help.............
Answers
Tanisha cut a triangle with angles of 34°, 67°, and 79° in three pieces so that each piece had one angle of the triangle. She arranged the three angles connected together such that the three corners of the triangle were all touching. What type of angle was formed and what does this show?