Answer:
The correct answer is B. : 10 sq mm = 0.1 sq cm
Step-by-step explanation:
It is just a matter of changing the units. The equivalence we need to know is 1cm = 10 mm. Also, we need to have in mind that we can write 10 sq mm as 10 mm*mm, because : 10 sq mm = 10 mm² = 10 mm*mm
Now we multiply two times by the fraction (1cm / 10 mm), which does not alter our measurement because the fraction is the same as multiplying by 1.
10 sq mm = 10 mm* mm = (10 mm*mm)*(1 cm / 10 mm)*(1 cm / 10 mm) = (10 mm*mm*cm*cm/ 10*10 mm*mm) =10/100 cm*cm = 0.1 cm² = 0.1 sq cm
Therefore, we have the equivalency : 10 sq mm = 0.1 sq cm
4. Mary is reviewing her algebra quiz. She has determined that one of her solutions is incorrect. Which one is it? A. 2x + 5 (x-1) 9, X 2 B. p - 3(p-5) 10, p 2.5 C. 4 y +3 y 28, y 4 E. t - 2t - 3t 32, t 8
Answer:
E. t - 2t - 3t =32, t=8
Step-by-step explanation:
A. 2x + 5 (x-1)= 9, X= 2 it's ok because
[tex]2x+5(x-1)=9\\ 2x+5x-5=9\\ 7x=14\\ x=2[/tex]
B. p - 3(p-5)= 10, p= 2.5 it's ok as well because
[tex]p-3(p-5)=10\\ p-3p+15=10\\ -2p=-5\\ p=\frac{-5}{-2} \\ p=2.5[/tex]
C. 4 y +3 y=28, y=4 also it's ok because
[tex]4y+3y=28\\ 7y=28\\ y=\frac{28}{7} \\ y=4[/tex]
E. t - 2t - 3t =32, t=8 it's not ok because
[tex]t-2t-3t=32\\-4t=32\\t=\frac{32}{-4} \\t=-8[/tex]
a (-) is missing!
Final answer:
After reviewing the solutions provided by Mary, we can conclude that the incorrect solution is Option E, where the solution should be t = -8 instead of the provided answer t = 8.
Explanation:
Mary is reviewing her algebra quiz and needs to determine which one of her solutions is incorrect. We must go through each option and verify the results given.
Option A
2x + 5(x - 1) = 9, solution X = 2.
First, expand the equation: 2x + 5x - 5 = 9.
Then combine like terms: 7x - 5 = 9.
Next, add 5 to both sides: 7x = 14.
Finally, divide by 7: x = 2.
This solution is correct.
Option B
p - 3(p - 5) = 10, solution p = 2.5.
First, expand the equation: p - 3p + 15 = 10.
Then combine like terms: -2p + 15 = 10.
Next, subtract 15 from both sides: -2p = -5.
Finally, divide by -2: p = 2.5.
This solution is correct.
Option C
4y + 3y = 28, solution y = 4.
First, combine like terms: 7y = 28.
Then, divide by 7: y = 4.
This solution is correct.
Option E
t - 2t - 3t = 32, solution t = 8.
First, combine like terms: -4t = 32.
Next, divide by -4: t = -8.
This solution is incorrect because the given answer is t = 8. The correct answer should be t = -8.
Ethanol fuel mixtures have "E" numbers that indicate the percentage of ethanol in the mixture by volume. For example, E10 is a mixture of 10% ethanol and 90% gasoline. How much E5 should be mixed with 6000 gal of E10 to make an E9 mixture?
To figure out how much E5 should be mixed with 6000 gal of E10 to make an E9 mixture, we need to use a weighted average equation: 0.10*6000 + 0.05*X = 0.09 * (6000 + X). Solving this equation will provide the required amount of E5 gasoline.
Explanation:To solve this problem, we use a technique known as a weighted average. The weight is the number of gallons and the 'value' is the percentage of ethanol. We can formulate an equation using the principle that the sum of the ethanol in the initial gasolines will equal to the ethanol in the final mixture.
Let's denote by X the needed amount of E5 gasoline. Therefore, the total ethanol before mixing would be 0.10*6000 (from the E10 gasoline) + 0.05*X (from the E5 gasoline). After mixing, the total ethanol will be 0.09 (6000 + X).
Equating these two gives us 0.10*6000 + 0.05*X = 0.09 * (6000 + X). Solving this equation will give the needed amount of E5 gasoline.
Learn more about Weighted Average here:https://brainly.com/question/36895489
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To create an E9 mixture, you need to mix 1500 gallons of E5 with 6000 gallons of E10. This is calculated using the volume percentages of ethanol in each mixture and solving a linear equation. The final mixture will contain 9% ethanol.
To solve the problem of finding out how much E5 should be mixed with 6000 gallons of E10 to create an E9 mixture, we need to set up an equation based on the percentage volumes of ethanol.
Let's denote the amount of E5 to be added as x (in gallons).
E10 contains 10% ethanol and 90% gasoline. Therefore, in 6000 gallons of E10, the amount of ethanol is:
0.10 × 6000 = 600 gallons of ethanol.E5 includes 5% ethanol and 95% gasoline. Therefore, in x gallons of E5, the amount of ethanol is:
0.05 × x gallons of ethanol.We want to create an E9 mixture, which means the final mixture should contain 9% ethanol. The total volume of the final mixture will be:
6000 + x gallons.The amount of ethanol in the final mixture can be represented as:
(600 + 0.05x) gallons of ethanol.This amount should be 9% of the total final volume:
0.09 × (6000 + x) = 600 + 0.05xSolving for x:
0.09(6000 + x) = 600 + 0.05x540 + 0.09x = 600 + 0.05x0.09x - 0.05x = 600 - 5400.04x = 60x = 1500Therefore, you need to mix 1500 gallons of E5 with 6000 gallons of E10 to create an E9 mixture.
In how many ways can 3 girls divide 10 pennies if each must end up with at least one penny?
Answer:
36.
Step-by-step explanation:
We are asked to find the number of ways in which 3 girls can divide 10 pennies such that each must end up with at least one penny.
The selection can be done by selecting two dividing likes between the 10 pennies such that the set is divided into three parts.
Since each girl must have one penny, so no girl can have 0 penny. So the dividing like cannot be placed at end points, beginning and at the end. Therefore, we are left with 9 positions.
Now we need to find number of ways to select two positions out of the 9 positions that is C(9,2).
[tex]_{2}^{9}\textrm{C}=\frac{9!}{7!*2!}=\frac{9*8*7!}{7!*2*1}=\frac{9*8}{2}=9*4=36[/tex]
Therefore, there are 36 ways to divide 10 pennies between 3 girls.
Solve the differential equation t dy/dt + dy/dt = te ^y
The answer is
t dy / dt + dy / dt = te ^ y
I apply common factor 1/dt*(tdy+dy)=te^y
I pass "dt" tdy+dy=(te^y)*dt
I apply common factor (t+1)*dy=(te^y)*dt
I pass "e^y" (1/e^y)dy=((t+1)*t)*dt
I apply integrals ∫ (1/e^y)dy= ∫ (t^2+t)*dt
by property of integrals ∫ (1/e^y)dy= ∫ (e^-y)dy
∫ (e^-y)dy= ∫ (t^2+t)*dt
I apply integrals
-e^-y=(t^3/3)+(t^2/2)+C
I apply natural logarithm to eliminate "e"
-ln (e^-y)=-ln(t^3/3)+(t^2/2)+C
y=ln(t^3/3)+(t^2/2)+C
The general solution of the differential equation is
[tex]y = - ln( - ln |t + 1| + c)[/tex]
How to find the particular solution.
Given differential equation
[tex]t \frac{dy}{dt} + \frac{dy}{dt} = t {e}^{y} [/tex]
we can rearrange it as:
[tex](t + 1) \frac{dy}{dt} = t {e}^{y} [/tex]
Now, we can separate variables:
[tex] \frac{dy}{t {e}^{y} } = \frac{dt}{t + 1} [/tex]
Integrating both sides
[tex]∫ \frac{1}{t {e}^{y} } dy = ∫ \frac{1}{t + 1} dt[/tex]
[tex] - {e}^{ - y} = ln |t + 1| + c[/tex]
[tex] {e}^{ - y} = ln |t + 1| + c[/tex]
[tex]y = - ln( - ln |t + 1| + c[/tex]
Where C1 is the constant of integration.
Karen is planning to drive 848 miles on a road trip. If she drives 212 miles a day, how many days will it take to complete the trip? # days License Points possible: 2
Answer: It will take 4 days to complete the whole trip.
Step-by-step explanation:
Given : Karen is planning to drive 848 miles on a road trip.
The number of miles she travels in a day = 212 miles
Then, the number of days taken to complete the whole trip will be [Divide 848 by 212]:_
[tex]\dfrac{848}{212}=4[/tex]
Hence, it will take 4 days to complete the whole trip.
Final answer:
To find the number of days Karen needs to complete her 848-mile road trip at a rate of 212 miles per day, divide the total miles by the daily miles, which results in 4 days.
Explanation:
The student's question asks: Karen is planning to drive 848 miles on a road trip. If she drives 212 miles a day, how many days will it take to complete the trip? This is a simple division problem in mathematics. To find out how many days it will take for Karen to complete the trip, you divide the total miles of the trip (848 miles) by the number of miles she can drive in a day (212 miles/day).
So the calculation would be: 848 miles ÷ 212 miles/day = 4 day
Therefore, it will take Karen 4 days to complete her road trip if she drives 212 miles each day.
Suppose that a worker in Country A can make either 10 iPods or 5 tablets each year. Country A has 100 workers. Suppose a worker in Country B can make either 2 iPods or 10 tablets each year. Country B has 200 workers. A bundle of goods that Country A could not make would be:
Answer:
A bundle of goods that Country A has are 1000 iPods and 500 Tablets.
Step-by-step explanation:
Country A can produce either iPods and Tablets.
Total man power Country A has 100 workers.
As per information in the question w
100 worker make either ipod or 100 worker make tablets
1 worker can produce 10 iPods.
Therefore
100 workers can produce (100 * 10) = 1000 iPods.
on the other hand
1 worker can produce 5 tablets.
Therefore
100 workers can produce (100 * 5)= 500 Tablets
Thus,
A bundle of goods that Country A has are 1000 iPods and 500 Tablets.
Find the point 3x - 7x₂ =5. of the lines x, - 5x2 = 1 and of intersection
Answer:
([tex]\frac{9}{4},\frac{1}{4})[/tex]
Step-by-step explanation:
We are given that two lines
[tex]x_1-7x_2=5[/tex] and [tex]x_1-5x_2=1[/tex]
We have to find the intersection point of two lines
Let [tex]3x_1-7x_2=5[/tex] (equation 1)
[tex]x_1-5x_2=1[/tex] (Equation 2)
Multiply equation 2 by 3 then subtract from equation 1
[tex]-7x_2+15x_2=5-3[/tex]
[tex]8x_2=2[/tex]
[tex]x_2=\frac{2}{8}=\frac{1}{4}[/tex]
Substitute [tex] x_2=\frac{1}{4}[/tex] in the equation 1
Then, we get
[tex]3x_1-7\frac{1}{4}=5[/tex]
[tex]3x_1-\frac{7}{4}=5[/tex]
[tex]3x_1=5+\frac{7}{4}=\frac{20+7}{4}=\frac{27}{4}[/tex]
[tex]x_1=\frac{27}{4\times 3}=\frac{9}{4}[/tex]
Hence, the intersection point of two given lines is ([tex]\frac{9}{4},\frac{1}{4})[/tex]
What is the quotient of the fractions below? 4/7 + 7/5
Answer:
(4/7) ÷ (7/5) = 20/49
Step-by-step explanation:
You show a sum, not a quotient.
Perhaps you want the quotient ...
(4/7) ÷ (7/5) = (4/7)×(5/7) = (4·5)/(7·7) = 20/49
xy′ = √(1 − y2 ), y(1) = 0
Answer:
The particular solution is [tex]y=\sin (\ln|x|)[/tex] .
Step-by-step explanation:
The given differential equation is
[tex]xy'=\sqrt {1-y^2}[/tex]
It can be written as
[tex]x\frac{dy}{dx}=\sqrt {1-y^2}[/tex]
Use variable separable method to solve the above equation.
[tex]\frac{dy}{\sqrt {1-y^2}}=\frac{1}{x}dx[/tex]
Integrate both sides.
[tex]\int \frac{dy}{\sqrt {1-y^2}}=\int \frac{1}{x}dx[/tex]
[tex]\sin^{-1} y=\ln|x|+C[/tex] .... (1)
It is given that y(1)=0. It means y=0 at x=1.
[tex]\sin (0)=\ln|1|+C[/tex]
[tex]0=0+C[/tex]
[tex]0=C[/tex]
The value of constant is 0.
Substitute C=0 in equation (1) to find The required equation.
[tex]\sin^{-1} y=\ln|x|+0[/tex]
Taking sin both sides.
[tex]y=\sin (\ln|x|)[/tex]
Therefore the particular solution is [tex]y=\sin (\ln|x|)[/tex] .
Senior management of a consulting services firm is concerned about a growing decline in the firm’s weekly number of billable hours. The firm expects each professional employee to spend at least 40 hours per week on work. In an effort to understand this problem better, management would like to estimate the standard deviation of the number of hours their employees spend on work-related activities in a typical week. Rather than reviewing the records of all the firm’s full-time employees, the management randomly selected a sample of size 51 from the available frame. The sample mean and sample standard deviations were 48.5 and 7.5 hours, respectively. Construct a 99% confidence interval for the standard deviation of the number of hours this firm’s employees spend on work-related activities in a typical week
Answer: (45.79, 51.21)
Step-by-step explanation:
Given : Significance level : [tex]\alpha: 1-0.99=0.01[/tex]
Sample size : n= 51 , which is a large sample (n>30), so we use z-test.
Critical value: [tex]z_{\alpha/2}=2.576[/tex]
Sample mean : [tex]\overline{x}= 48.5\text{ hours}[/tex]
Standard deviation : [tex]\sigma=7.5\text{ hours}[/tex]
The confidence interval for population means is given by :-
[tex]\overline{x}\pm z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}[/tex]
i.e. [tex]48.5\pm(2.576)\dfrac{7.5}{\sqrt{51}}[/tex]
i.e.[tex]48.5\pm2.70534112234\\\\\approx48.5\pm2.71\\\\=(48.5-2.71, 48.5+2.71)=(45.79, 51.21)[/tex]
Hence, 99% confidence interval for the standard deviation of the number of hours this firm’s employees spend on work-related activities in a typical week = (45.79, 51.21)
Final answer:
To construct a 99% confidence interval for the standard deviation of firm's employees' work hours, follow the steps to calculate the interval between 6.26 and 9.08 hours.
Explanation:
Confidence Interval Calculation:
Calculate the degrees of freedom (df) using the formula df = n - 1. For a sample size of 51, df = 51 - 1 = 50.
Determine the critical values from the chi-squared distribution for a 99% confidence level with 50 degrees of freedom. These critical values are 30.984 and 73.361.
Next, calculate the confidence interval for the standard deviation using the formula CI = sqrt((n-1) ×s² / chi-squared upper) to sqrt((n-1) ×s² / chi-squared lower), which results in 6.26 to 9.08 hours.
If you roll one die and flip one coin, what is the probability of rolling a 2 and flipping a head? Why?
Answer:
Probability of rolling a 2 and flipping a head will be [tex]\frac{1}{12}[/tex]
Step-by-step explanation:
If we roll one die then probability to get any one side is [tex]\frac{1}{6}[/tex]
Therefore, probability to get 2 by rolling the die will be P(A) = [tex]\frac{1}{6}[/tex]
Now we flip a coin then getting head or tale probability is [tex]\frac{1}{2}[/tex]
Or probability to get head by flipping the coin P(B) = [tex]\frac{1}{2}[/tex]
Probability of happening both the events (rolling a 2 and flipping a head) will be denoted by
P(A∩B) = P(A)×P(B)
= [tex]\frac{1}{6}\times \frac{1}{2}[/tex]
= [tex]\frac{1}{12}[/tex]
Therefore, probability of rolling a 2 and flipping a head will be [tex]\frac{1}{12}[/tex]
Answer:
The probability of rolling a 2 and flipping a head is [tex]\frac{1}{12}[/tex]
Step-by-step explanation:
Notice that rolling a die and flipping a coin are two independent events this means that the probability that one event occurs in no way affects the probability of the other event occurring. When we determine the probability of two independent events we multiply the probability of the first event by the probability of the second event.
[tex]P(X and \:Y) =P(X) \cdot P(Y)[/tex]
When you roll a die there six outcomes from 1 to 6 and when you flip a coin are two possible outcomes (heads or tails).
We know that the probability of an event is
[tex]P=\frac{the \:number \:of \:wanted\:outcomes}{the \:number \:of \:possible \:outcomes}[/tex]
So the probability of rolling a 2 and flipping a head is
[tex]P(2 \:and \:H) = P(2) \cdot P(H)\\P(2 \:and \:H) = \frac{1}{2} \cdot \frac{1}{6}\\P(2 \:and \:H) = \frac{1}{12}[/tex]
Show that 2^2n-1 +1 is divisible by 3 for all n > 1.
Answer:
The proof makes use of congruences as follows:
Step-by-step explanation:
We can prove this result using congruences module 3. First of all we shall show that
[tex]2^{2n-1}\equiv 2 \pmod{3}[/tex] for all [tex]n\in \mathbb{N}[/tex]. By induction we have
[tex]n=2[/tex]. For [tex]n=2[/tex] we have [tex]2^{4-1}=8\equiv 2 \pmod{3}[/tex]Suppose that the statement is true for [tex]n=k[/tex] and let's prove that it is also true for [tex]n=k+1[/tex]. In fact, [tex]2^{2(k+1)-1}=2^{2k-1+2}=2^{2k-1}2^{2}\equiv 2\cdot 2^{2}\equiv 8 \equiv 2 \pmod{3}[/tex]Then induction we proved that [tex]2^{2n-1}\equiv 2 \pmod{3}[/tex] for all [tex]n>1[/tex]. Then
[tex]2^{2n-1}+1\equiv 2+1\equiv 3\equiv 0 \pmod{3}[/tex]
From here we conclude that the expression [tex]2^{2n-1}+1[/tex] is divisible by 3.
For the following linear system, put the augmented coefficient matrix into reduced row-echelon form.
2x1 + 3x2 − x3 = 14
x1 + 2x2 + x3 = 4
5x1 + 9x2 + 2x3 = 7
1 0 -5 16
0 1 3 -6
0 0 0 -19
Incorrect
Answer:
The reduced row-echelon form of the linear system is [tex]\left[\begin{array}{cccc}1&0&-5&0\\0&1&3&0\\0&0&0&1\end{array}\right][/tex]
Step-by-step explanation:
We will solve the original system of linear equations by performing a sequence of the following elementary row operations on the augmented matrix:
Interchange two rowsMultiply one row by a nonzero numberAdd a multiple of one row to a different rowTo find the reduced row-echelon form of this augmented matrix
[tex]\left[\begin{array}{cccc}2&3&-1&14\\1&2&1&4\\5&9&2&7\end{array}\right][/tex]
You need to follow these steps:
Divide row 1 by 2 [tex]\left(R_1=\frac{R_1}{2}\right)[/tex][tex]\left[\begin{array}{cccc}1&3/2&-1/2&7\\1&2&1&4\\5&9&2&7\end{array}\right][/tex]
Subtract row 1 from row 2 [tex]\left(R_2=R_2-R_1\right)[/tex][tex]\left[\begin{array}{cccc}1&3/2&-1/2&7\\0&1/2&3/2&-3\\5&9&2&7\end{array}\right][/tex]
Subtract row 1 multiplied by 5 from row 3 [tex]\left(R_3=R_3-\left(5\right)R_1\right)[/tex][tex]\left[\begin{array}{cccc}1&3/2&-1/2&7\\0&1/2&3/2&-3\\0&3/9&9/2&-28\end{array}\right][/tex]
Subtract row 2 multiplied by 3 from row 1 [tex]\left(R_1=R_1-\left(3\right)R_2\right)[/tex][tex]\left[\begin{array}{cccc}1&0&-5&16\\0&1/2&3/2&-3\\0&3/9&9/2&-28\end{array}\right][/tex]
Subtract row 2 multiplied by 3 from row 3 [tex]\left(R_3=R_3-\left(3\right)R_2\right)[/tex][tex]\left[\begin{array}{cccc}1&0&-5&16\\0&1/2&3/2&-3\\0&0&0&-19\end{array}\right][/tex]
Multiply row 2 by 2 [tex]\left(R_2=\left(2\right)R_2\right)[/tex][tex]\left[\begin{array}{cccc}1&0&-5&16\\0&2&3&-6\\0&0&0&-19\end{array}\right][/tex]
Divide row 3 by −19 [tex]\left(R_3=\frac{R_3}{-19}\right)[/tex][tex]\left[\begin{array}{cccc}1&0&-5&16\\0&2&3&-6\\0&0&0&1\end{array}\right][/tex]
Subtract row 3 multiplied by 16 from row 1 [tex]\left(R_1=R_1-\left(16\right)R_3\right)[/tex][tex]\left[\begin{array}{cccc}1&0&-5&0\\0&1&3&-6\\0&0&0&1\end{array}\right][/tex]
Add row 3 multiplied by 6 to row 2 [tex]\left(R_2=R_2+\left(6\right)R_3\right)[/tex][tex]\left[\begin{array}{cccc}1&0&-5&0\\0&1&3&0\\0&0&0&1\end{array}\right][/tex]
Water leaks from a vertical cylindrical tank through a small hole in its base at a volumetric rate proportional to the square root of the volume of water remaining. The tank initially contains 300 liters and 22 liters leak out during the first day A. When will the tank be half empty?t16 day (include units.) B. How much water will remain in the tank after 5 days? volume 198 L (include units)
Answer:
After 7.84 the tank be half empty. The water remain in the tank after 5 days is 198.401 L.
Step-by-step explanation:
Consider the provided information.
It is given that a small hole in its base at a rate proportional to the square root of the volume of water remaining. The tank initially contains 300 liters and 22 liters leak out during the first day.
The rate of water leak can be written as:
[tex]\frac{dV}{dt}\propto \sqrt{V}[/tex]
Let k be the constant of proportionality.
[tex]\frac{dV}{dt}=k \sqrt{V}[/tex]
Integrate both the sides as shown:
[tex]\frac{dV}{\sqrt{V}}=k dt\\\int\frac{dV}{\sqrt{V}}=\int k dt\\2\sqrt{V} =kt+c[/tex]
Since for t=0 the volume was 300.
[tex]2\sqrt{300} =k(0)+c\\20\sqrt{3} =c\\c=34.641[/tex]
Now substitute the value of c in above equation.
[tex]2\sqrt{V} =kt+34.641[/tex]
22 liters leak out during the first day, thus now the remaining volume is 300-22=278 liters.
[tex]2\sqrt{278} =k(1)+34.641\\33.347 =k+34.641\\k=33.347 -34.641\\k=-1.294[/tex]
Thus, the required equation is:[tex]2\sqrt{V} =-1.294t+34.641[/tex]
Part (A) When will the tank be half empty.
Substitute v=150 liters for half empty in above equation.
[tex]2\sqrt{150} =-1.294t+34.641[/tex]
[tex]24.495 =-1.294t+34.641[/tex]
[tex]-10.146 =-1.294t[/tex]
[tex]t=7.84[/tex]
Hence, after 7.84 the tank be half empty.
Part (B) How much water will remain in the tank after 5 days.
Substitute the value of t=5 in [tex]2\sqrt{V} =-1.294t+34.641[/tex]
[tex]2\sqrt{V} =-1.294(5)+34.641[/tex]
[tex]2\sqrt{V} =28.171[/tex]
[tex]\sqrt{V} =14.0855[/tex]
[tex]V =198.401[/tex]
Hence, the water remain in the tank after 5 days is 198.401 L.
The correct answer is A) The tank will be half empty in 16 days, B) The remaining volume after 5 days will be 198 L.
A) To find when the tank will be half empty, we need to solve the differential equation that models the rate of change of the volume of water in the tank.
Let V(t) be the volume of water remaining in the tank at time t.
The rate of change of the volume is proportional to the square root of the volume:
dV/dt = -k√V
where k is a constant that can be determined from the given information.
We know that V(0) = 300 L and V(1) = 300 - 22 = 278 L.
Substituting these values, we get:
k = 22 / √300 = 4
Solving the differential equation with the initial condition V(0) = 300, we get:
[tex]V(t) = 300^_{(1/2)}$-2t^_2[/tex]
Setting V(t) = 150 L (half of the initial volume), we get:
t = 16 days
B) To find the volume remaining after 5 days, we substitute t = 5 in the solution:
[tex]V(5) = (300^_(1/2)} - 2(5))^2 = (\sqrt{300} - 10)^2 = 198 L[/tex]
The rate of change of the volume is proportional to the square root of the volume, which leads to a separable differential equation. By using the given information to determine the constant of proportionality, we can solve the differential equation and find the time when the volume is halved. Substituting the desired time into the solution gives the remaining volume after that time.
Write down the explicit solution for each of the following: a) x’=t–sin(t); x(0)=1
b) x’+2x=4; x(0)=5
c) x’’+4x=0; x(0)=0; x’(0)=1
Answer:
a) x=(t^2)/2+cos(t), b) x=2+3e^(-2t), c) x=(1/2)sin(2t)
Step-by-step explanation:
Let's solve by separating variables:
[tex]x'=\frac{dx}{dt}[/tex]
a) x’=t–sin(t), x(0)=1
[tex]dx=(t-sint)dt[/tex]
Apply integral both sides:
[tex]\int {} \, dx=\int {(t-sint)} \, dt\\\\x=\frac{t^2}{2}+cost +k[/tex]
where k is a constant due to integration. With x(0)=1, substitute:
[tex]1=0+cos0+k\\\\1=1+k\\k=0[/tex]
Finally:
[tex]x=\frac{t^2}{2} +cos(t)[/tex]
b) x’+2x=4; x(0)=5
[tex]dx=(4-2x)dt\\\\\frac{dx}{4-2x}=dt \\\\\int {\frac{dx}{4-2x}}= \int {dt}\\[/tex]
Completing the integral:
[tex]-\frac{1}{2} \int{\frac{(-2)dx}{4-2x}}= \int {dt}[/tex]
Solving the operator:
[tex]-\frac{1}{2}ln(4-2x)=t+k[/tex]
Using algebra, it becomes explicit:
[tex]x=2+ke^{-2t}[/tex]
With x(0)=5, substitute:
[tex]5=2+ke^{-2(0)}=2+k(1)\\\\k=3[/tex]
Finally:
[tex]x=2+3e^{-2t}[/tex]
c) x’’+4x=0; x(0)=0; x’(0)=1
Let [tex]x=e^{mt}[/tex] be the solution for the equation, then:
[tex]x'=me^{mt}\\x''=m^{2}e^{mt}[/tex]
Substituting these equations in c)
[tex]m^{2}e^{mt}+4(e^{mt})=0\\\\m^{2}+4=0\\\\m^{2}=-4\\\\m=2i[/tex]
This becomes the solution m=α±βi where α=0 and β=2
[tex]x=e^{\alpha t}[Asin\beta t+Bcos\beta t]\\\\x=e^{0}[Asin((2)t)+Bcos((2)t)]\\\\x=Asin((2)t)+Bcos((2)t)[/tex]
Where A and B are constants. With x(0)=0; x’(0)=1:
[tex]x=Asin(2t)+Bcos(2t)\\\\x'=2Acos(2t)-2Bsin(2t)\\\\0=Asin(2(0))+Bcos(2(0))\\\\0=0+B(1)\\\\B=0\\\\1=2Acos(2(0))\\\\1=2A\\\\A=\frac{1}{2}[/tex]
Finally:
[tex]x=\frac{1}{2} sin(2t)[/tex]
What is the equation of the line that has a slope of 3 and goes through the point (-3,-5)?
Answer:
Step-by-step explanation:
Let's remember the equation of a line:
y = mx + b
m: slope (we know it's 3)
b: the y-intercept
So far, we have y = 3x + b, now we need to find b.
Replacing y and x for the given points (-3,-5):
-5 = 3*(-3) + b
-5 = -9 + b
b = -5 + 9
b = 4
The equation of the line that passes through the point (-3,-5) with a slope of 3 is y = 3x + 4
Answer:
y = 3x + 4
Step-by-step explanation:
(1/2) (10t) = 50,000
Answer: The value of t become 10000.
Step-by-step explanation:
Since we have given that
[tex]\dfrac{1}{2}\times 10t=50000[/tex]
We need to find the value of t from the above expression:
first we divide 10t by 2 to get 50000.
[tex]\dfrac{10t}{2}=50000\\\\5t=50000\\\\t=\dfrac{50000}{5}\\\\t=10000[/tex]
Hence, the value of t become 10000.
A breath analyzer, used by the police to test whether drivers exceed the legal limit for blood alcohol percentage while driving, is known to satisfy P ( A | B ) = P ( A c | B c ) = p where A is the event "breath analyzer indicates that legal limit is exceeded" and B is the event "driver's blood alcohol percentage exceeds legal limit." On Saturday night, about 5% of all drivers are known to exceed the limit. If we want P ( B | A ) to equal 0.9, what value of p should we use, rounded to 4 decimal places? Group of answer choices
To find the value of 'p' that satisfies the equation, we use Bayes' theorem and substitute the given values into the equation.
Explanation:In this question, we need to find the value of p which satisfies the equation P(A|B) = P(A' | B'), where A is the event 'breath analyser indicates that legal limit is exceeded' and B is the event 'driver's blood alcohol percentage exceeds legal limit.'
We are given that approximately 5% of all drivers exceed the legal limit for blood alcohol percentage while driving, which means P(B) = 0.05.We are also given that we want P(B|A) = 0.9. Using Bayes' theorem, we can write:Therefore, the value of p that should be used, rounded to 4 decimal places, can be calculated using the above steps.
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A web page design firm has two designs for an online hardware store. To determine which is the more effective design, the firm uses one page when on a mobile platform and a second page when on a desktop. For each visit, the firm records the amount spent by the visitor and the amount of time visiting the site. Complete parts (a) through (c) below (a) What is the explanatory variable in this study? Is it qualitative or quantitative? O A. The explanatory variable is the amount spent by the visitor. This explanatory variable is quantitative. O B. The explanatory variable is the web page design. The explanatory variable is qualitative. ° C. The explanatory variable is the amount of time visiting the site. This explanatory variable is quantitative O D. The explanatory variable is the age group. This explanatory variable is qualitative. (b) What are the two response variables? For each response variable, state whether it is qualitative or quantitative. Select all that apply. □ A- One response variable is the amount spent by the visitor. This response variable is quantitative. B. One response variable is the web page design. This response variable is qualitative. C. One response variable is the age group. This response variable is qualitative D. One response variable is the amount of time visiting the site. This response variable is quantitative. (c) Explain how confounding might be an issue with this study. Choose the correct answer below d A. Since there are two response variables in this study, they may interact, causing the results to be misleading or incorrect. B. Since there is only one explanatory variable, and two response variables, the explanatory variable will not affect both response variables equally, resulting in misleading results O C. Since the designs are being tested with two different age groups, preferences depending on the age group may affect the response variables for those groups. O D. Since the variables are not all quantitative or all qualitative, they may interact, causing the results to be misleading or incorrect.
The explanatory variable is the web page design. The response variables are the amount spent by the visitor and the amount of time visiting the site. Confounding could be an issue if other factors are not accounted for.
Explanation:(a) The explanatory variable in this study is the web page design. It is a qualitative variable because it represents different design options.
(b) The two response variables are the amount spent by the visitor and the amount of time visiting the site. The amount spent is a quantitative variable, while the amount of time is also quantitative.
(c) Confounding could be an issue in this study if there are other factors that could affect the response variables and are not accounted for. For example, if there are differences in the age groups of the visitors using mobile and desktop platforms, their preferences may affect the response variables.
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If there are 2.54 centimeters in 1 inch, how many centimeters are in 500 inches
Answer: There are 196.85 cm in 500 inches.
Step-by-step explanation:
Since we have given that
1 inch = 2.54 centimeters
We need to find the number of centimeters are in 500 inches.
Since 1 inch = 2.54 centimeters
1 inch = [tex]\dfrac{1}{2.54}\ cm[/tex]
500 inches = [tex]\dfrac{500}{2.54}=196.85\ cm[/tex]
Hence, There are 196.85 cm in 500 inches.
To convert 500 inches to centimeters, use the conversion factor 2.54 cm = 1 inch. Cross-multiply and solve for x to find that there are approximately 1270 centimeters in 500 inches.
Explanation:To convert inches to centimeters, we use the conversion factor 2.54 cm = 1 inch. In this case, we have 500 inches, so we can set up the following proportion:
We can cross-multiply and solve for x:
Calculating x, we find that there are approximately 1270 centimeters in 500 inches.
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ANSWER PLEASE ITS AN EMERGENCY. COULD SOMEONE ALSO HELP WITH OTHER QUESTIONS TOO?ILL GIVE BRAINLIEST
Angle G and Angle K are complementary angles.
The measure of Angle K is 5 times the measure of Angle G.
Find the measures of Angle G and Angle K.
Measure of Angle G =
degrees.
Measure of Angle K =
degrees.
Answer:
angle k:75
angle g:15
Step-by-step explanation:
What does frequency refer to as it pertains to a frequency histogram? а. Proportion b. Count C. Mean d. Variance
Answer:
(a) PROPORTION.
Step-by-step explanation:
A relative frequency histogram has bars whose height is equal to either the proportion of cases that are between the upper and lower bounds of the bar.
The relative frequency in a relative frequency histogram refers to PROPORTION.
A relative frequency histogram is a graph that has the same shape and the same horizontal scale as the corresponding frequency histogram. The difference is that the vertical scale measures the relative frequencies (percentages or proportions).
Hence, option (a) is right option.
Team A and Team B play each other in a best out of 7 tournament. So the team that wins 4 games first wins the tournament. How many possible sequences are there for team A to win? Examples: AAAA, AAABA, AABAA, etc.
Answer: There 64 possible sequences in which team A wins
Step-by-step explanation:
Hi!
The sequences in which team A wins, are the ones with at least 4 A's.
Sequences with 4, 5, 6 or 7 A's. To calculate how many of each type exist, we use the formula of combinations. If you select K objects from a set of N objects, there are C(N, K) possibilities, give by the formula:
[tex]C(N,K) = \frac{N!}{K! (N-K)!)}[/tex]
Then the total M number of sequences in which team A wins is:
[tex]M = C(7,7) + C(7,6) + C(7,5) + C(7,4) = 1 + 7 +21+35 = 64[/tex]
Which of the following completes the statement?
In the number 45,569, the 5 in the hundreds place is ______ the 5 to its left.
A. the same value as
B. 1/10 the value of
C. 10 times the value of
D. 100 times the value of
Answer:
B. 1/10 the value of
Step-by-step explanation:
In our base-10 place-value number system, the place value of a number is multiplied by 10 when it moves 1 place to the left. It is multiplied by 1/10 when it moves 1 place to the right.
A digit has 1/10 the value of the same digit one place to its left.
Answer: B
Step-by-step explanation:
The amount of garbage, G, in tons per week, produced by a city with population p, measured in thousands of people, is given by G = f ( p ) The town of Tola has a population of 50,000 and produces 14 tons of garbage each week. Express this information in terms of the function f
Answer: [tex]G=f(50)=14[/tex]
Step-by-step explanation:
Given : The amount of garbage, G, in tons per week, produced by a city with population p, measured in thousands of people, is given by G = f(p).
Also, The town of Tola has a population of 50,000 and produces 14 tons of garbage each week.
Then, the expression in terms of the function G will be :-
[tex]G=f(50)=14[/tex]
A tank initially contains 60 gallons of brine, with 30 pounds of salt in solution. Pure water runs into the tank at 3 gallons per minute and the well-stirred solution runs out at the same rate. How long will it be until there are 23 pounds of salt in the tank? Answer: the amount of time until 23 pounds of salt remain in the tank is minutes.
Answer:
the amount of time until 23 pounds of salt remain in the tank is 0.088 minutes.
Step-by-step explanation:
The variation of the concentration of salt can be expressed as:
[tex]\frac{dC}{dt}=Ci*Qi-Co*Qo[/tex]
being
C1: the concentration of salt in the inflow
Qi: the flow entering the tank
C2: the concentration leaving the tank (the same concentration that is in every part of the tank at that moment)
Qo: the flow going out of the tank.
With no salt in the inflow (C1=0), the equation can be reduced to
[tex]\frac{dC}{dt}=-Co*Qo[/tex]
Rearranging the equation, it becomes
[tex]\frac{dC}{C}=-Qo*dt[/tex]
Integrating both sides
[tex]\int\frac{dC}{C}=\int-Qo*dt\\ln(\abs{C})+x1=-Qo*t+x2\\ln(\abs{C})=-Qo*t+x\\C=exp^{-Qo*t+x}[/tex]
It is known that the concentration at t=0 is 30 pounds in 60 gallons, so C(0) is 0.5 pounds/gallon.
[tex]C(0)=exp^{-Qo*0+x}=0.5\\exp^{x} =0.5\\x=ln(0.5)=-0.693\\[/tex]
The final equation for the concentration of salt at any given time is
[tex]C=exp^{-3*t-0.693}[/tex]
To answer how long it will be until there are 23 pounds of salt in the tank, we can use the last equation:
[tex]C=exp^{-3*t-0.693}\\(23/60)=exp^{-3*t-0.693}\\ln(23/60)=-3*t-0.693\\t=-\frac{ln(23/60)+0.693}{3}=-\frac{-0.959+0.693}{3}= -\frac{-0.266}{3}=0.088[/tex]
Aspirin tablets generally contain 325 mg of aspirin. How many such tablets may be prepared from 5 kg of aspirin?
Answer:
15,384 such tablets may be prepared from 5 kg of aspirin
Step-by-step explanation:
The problem states that aspirin tablets generally contain 325 mg of aspirin. And asks how many such tablets may be prepared from 5 kg of aspirin.
Since the problem measures the weight of a tablet in kg, the first step is the conversion of 325mg to kg.
Each kg has 1,000,000mg. So
1kg - 1,000,000mg
xkg - 325mg.
1,000,000x = 325
[tex]x = \frac{325}{1,000,000}[/tex]
x = 0.000325kg
Each tablet generally contains 0.000325kg of aspirin. How many such tablets may be prepared from 5 kg of aspirin?
1 tablet - 0.000325kg
x tablets - 5kg
0.000325x = 5
[tex]x = \frac{5}{0.000325}[/tex]
x = 15,384 tablets
15,384 such tablets may be prepared from 5 kg of aspirin
Around 15,385 tablets of aspirin can be created from 5 kilograms of aspirin, using the assumption that one tablet typically contains 325 milligrams of aspirin.
Explanation:This question requires a basic understanding of the conversion from kilograms to milligrams. 5 kilograms of aspirin is equal to 5,000,000 milligrams (as 1 kilogram = 1,000,000 milligrams). So, if one aspirin tablet contains 325 mg of aspirin, we can calculate the number of tablets from 5kg, or 5,000,000 mg, of aspirin by simply dividing the total milligrams of aspirin by the milligrams per tablet.
To calculate:
Number of tablets = Total mass / Mass per tablet
= 5,000,000 mg / 325 mg/tablet
So, roughly 15,385 tablets of aspirin can be produced from 5 kg of aspirin.
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Divide and simplify to the form a+bi. 5+6i 5+6i 6+ i (Simplify your answer. Type an integer or a fraction. Type your answer in the form a+bi.)
Answer:
[tex]-\frac{6}{37} + \frac{371}{37}i[/tex]
Step-by-step explanation:
We need to evaluate [tex]\frac{(5+6i)(5+6i)}{6+i}[/tex]
(5+6i)(5+6i) = (25 + 36i² + 60i) = (25 - 36 + 60i) = -11 + 60i
= [tex]\frac{-11+60i}{6+i}[/tex]
Now we rationalize the denominator.
Now, multiplying both the numerator and denominator by (6-i)
[tex]\frac{-66 + 11i + 360i - 60i^2}{36 - i^2} = \frac{-66 + 60 + 371i}{37} = \frac{-6 + 371i}{37}[/tex]
= [tex]-\frac{6}{37} + \frac{371}{37}i[/tex]
Formula used:
(a+b)² = a² + b² + 2ab
i² = -1
Please convert these values from scientific notation into long hand notation. To receive credit, plea: values over 999; use 0.# for values less than one; keep the two significant figuares. 3.3x1010 5.4x10-6 7.2x10-1 4.5x100=
Answer:
The required answers are:
33,000,000,000
0.0000054
0.72
4.5
Step-by-step explanation:
Consider the provided information.
We can convert the scientific notation into long hand notation as shown:
If the exponent of 10 is a positive number then move the decimal point right as much as the exponent value.
If the exponent of 10 is a negative number then move the decimal point left as much as the exponent value.
Part (A) [tex]3.3\times 10^{10}[/tex]
Here the exponent of 10 is 10 which is a positive number, so move the decimal point right as shown:
[tex]3.3\times 10^{10}=33,000,000,000[/tex]
Part (B) [tex]5.4\times 10^{-6}[/tex]
Here the exponent of 10 is -6 which is a negative number, so move the decimal point left as shown:
[tex]5.4\times 10^{-6}=0.0000054[/tex]
Part (C) [tex]7.2\times 10^{-1}[/tex]
Here the exponent of 10 is -1 which is a negative number, so move the decimal point left as shown:
[tex]7.2\times 10^{-1}=0.72[/tex]
Part (D) [tex]4.5\times 10^0[/tex]
Here the exponent of 10 is 0. Use the property of exponent [tex]a^0=1[/tex]
[tex]4.5\times 10^0=4.5[/tex]
Ana and Bella are sisters.They are responsible for keeping the
family's driveway clear of snow.If they work together they can
complete the shoveling in 30 minutes.
One day Ana had to do the job alone.It took her 40 minutes.How long
would it have taken Bella to shovel the same amont alone?You can
assume that each girl always shovels at a constant speed.
Answer:
Bella can complete work alone in 120 minutes
Step-by-step explanation:
Ana and Bella together can complete the shoveling in 30 minutes.
They can do a part of work in 1 minute = [tex]\frac{1}{30}[/tex]
Ana can do work alone in 40 minutes
She can do part of work in 1 minute = [tex]\frac{1}{40}[/tex]
So, Bella can do a part of work in 1 minute = [tex]\frac{1}{30}-\frac{1}{40}[/tex]
= [tex]\frac{1}{120}[/tex]
Bella can do [tex]\frac{1}{120}[/tex] part of work in 1 minute
Bella can do whole work in minutes = 120
Hence Bella can complete work alone in 120 minutes