The vector pointing from (2, 1) to (1, 3) points in the same direction as the vector [tex]\vec u=(1,3)-(2,1)=(-1,2)[/tex]. The derivative of [tex]f[/tex] at (2, 1) in the direction of [tex]\vec u[/tex] is
[tex]D_{\vec u}f(2,1)=\nabla f(2,1)\cdot\dfrac{\vec u}{\|\vec u\|}[/tex]
We have
[tex]\|\vec u\|=\sqrt{(-1)^2+2^2}=\sqrt5[/tex]
Then
[tex]D_{\vec u}f(2,1)=(f_x(2,1),f_y(2,1))\cdot\dfrac{(-1,2)}{\sqrt5}=\dfrac{-f_x(2,1)+2f_y(2,1)}{\sqrt5}=-\dfrac2{\sqrt5}[/tex]
[tex]\implies f_x(2,1)-2f_y(2,1)=2[/tex]
The vector pointing from (2, 1) to (5, 5) has the same direction as the vector [tex]\vec v=(5,5)-(2,1)=(3,4)[/tex]. The derivative of [tex]f[/tex] at (2, 1) in the direction of [tex]\vec v[/tex] is
[tex]D_{\vec v}f(2,1)=\nabla f(2,1)\cdot\dfrac{\vec v}{\|\vec v\|}[/tex]
[tex]\|\vec v\|=\sqrt{3^2+4^2}=5[/tex]
so that
[tex](f_x(2,1),f_y(2,1))\cdot\dfrac{(3,4)}5=1[/tex]
[tex]\implies3f_x(2,1)+4f_y(2,1)=5[/tex]
Solving the remaining system gives [tex]f_x(2,1)=\dfrac95[/tex] and [tex]f_y(2,1)=-\dfrac1{10}[/tex].
1) Homer and Marge have purchased a home for $189 000. The real estate agent informs them that homes in their area have generally depreciated by 11% every six years. Based on this, how much should they be able to sell their home for in 15 years? (3 points)
Answer:
They should be able to sell their home for $149706.9
Step-by-step explanation:
Let's first understand the situation.
There is an initial value for the house which is $189000. However, this value varies every 6 years because of a 11% depreciation of the total value.
Because the depreciation is not executed during the 6 years in a constant way, but instead after the whole 6 years have passed, then we can calculate how many depreciations will be applied within the next 15 years:
total years/years needed for depreciation=15years/6years=2.5
The above means that only 2 depreciations are going to be applied. Remember that depreciation is only applied if the whole 6 years have passed.
Now, after the first 6 years the depreciation (D) is:
D = 0.11 * $189000 = $20790,
which means that the value of the house will be:
(initial value) - D = $189000 - $20790 = $168210
Now, after the following 6 years, first 12 years, the depreciation (D) is:
D = 0.11 * $168210 = $18503.1,
which means that the value of the house will be:
(initial value) - D = $168210 - $18503.1 = $149706.9
In conclusion, in 15 years from now, they should be able to sell their home for $149706.9
Write an equation of the horizontal asymptote for this function. Also, interpret what this asymptote means in the context of the problem (in terms of the fish population and the number of years since the fish were introduced into the lake.)
Answer:
Step-by-step explanation:
First, finding the horizontal asymptote:
[tex]\lim_{t \to \infty} = \frac{200+40t}{1+0.05t} = \frac{\frac{200}{t} 40 }{\frac{1}{t} 0.05} = 800[/tex]
In the context of the problem, the horizontal asymptote speaks about where the population of the fish is headed and capped.
Louisa ran at an average speed of five miles per hour along an entire circular park path. Calvin ran along the same path in the opposite direction at an average speed of six miles per hour. It took Calvin 30 minutes less than it took Louisa to run the full path once. How many miles did Louisa run when she completed one circular path?
Answer:
15 miles
Step-by-step explanation:
Let [tex]x[/tex] be the miles in the circular park path, [tex]t_{L}[/tex] the time Louisa takes to finish and [tex]t_{C}[/tex] the time Calvin takes to finish both in hours.
Then [tex]x[/tex], the longitude is equal to the velocity times the time used to finish. So
[tex]x=5t_{L}[/tex]
[tex]x=6t_{C}[/tex]
And the difference between Louisa's time and Calvin' time is 30 minutes, half an hour. So:
[tex]t_{C}=t_{L}-0.5[/tex]
Three equations, three unknowns, the system can be solved.
Equalizing the equation with x :
[tex]5t_{L}=6t_{C}[/tex]
In this last equation replace [tex]t_{C}[/tex] with the other equation and solve:
[tex]5t_{L}=6(t_{L}-0.5)\\ 5t_{L}=6t_{L}-3\\ 3=6t_{L}-5t_{L}\\ 3=t_{L}\\ t_{L}=3[/tex]
With Louisa's time find x:
[tex]x=5t_{L}\\ x=5(3)\\ x=15[/tex]
The length of the circular path that Louisa ran is 15 miles when she completed one circular path.
Let's denote the length of the circular path as L miles. Louisa's speed is 5 miles per hour, and Calvin's speed is 6 miles per hour.
We'll use the equation for time, which is time = distance / speed.
For Louisa:
[tex]t_L = \frac{L}{5}[/tex]
Where:
[tex]t_L[/tex] is the time it takes Louisa to complete the path.L is the length of the circular path.5 is Louisa's speed in miles per hour.For Calvin:
[tex]t_C = \frac{L}{6}[/tex]
Where:
[tex]t_C[/tex] is the time it takes Calvin to complete the path.L is the length of the circular path.6 is Calvin's speed in miles per hour.According to the problem, it took Calvin 30 minutes (0.5 hours) less than it took Louisa to run the full path:
[tex]t_L = t_C + 0.5[/tex]
Substitute the expressions for [tex]t_L[/tex] and [tex]t_C[/tex] into the equation:
[tex]\frac{L}{5} = \frac{L}{6} + 0.5[/tex]
To solve for L, first find a common denominator for the fractions.
The common denominator for 5 and 6 is 30.
Therefore:
[tex]\frac{6L}{30} = \frac{5L}{30} + 0.5[/tex]
Multiply everything by 30 to eliminate the denominators:
[tex]6L = 5L + 15[/tex]
Subtract 5L from both sides:
[tex]L = 15[/tex]
Solve the following equations: (a) x^11=13 mod 35 (b) x^5=3 mod 64
a.
[tex]x^{11}=13\pmod{35}\implies\begin{cases}x^{11}\equiv13\equiv3\pmod5\\x^{11}\equiv13\equiv6\pmod7\end{cases}[/tex]
By Fermat's little theorem, we have
[tex]x^{11}\equiv (x^5)^2x\equiv x^3\equiv3\pmod5[/tex]
[tex]x^{11}\equiv x^7x^4\equiv x^5\equiv6\pmod 7[/tex]
5 and 7 are both prime, so [tex]\varphi(5)=4[/tex] and [tex]\varphi(7)=6[/tex]. By Euler's theorem, we get
[tex]x^4\equiv1\pmod5\implies x\equiv3^{-1}\equiv2\pmod5[/tex]
[tex]x^6\equiv1\pmod7\impleis x\equiv6^{-1}\equiv6\pmod7[/tex]
Now we can use the Chinese remainder theorem to solve for [tex]x[/tex]. Start with
[tex]x=2\cdot7+5\cdot6[/tex]
Taken mod 5, the second term vanishes and [tex]14\equiv4\pmod5[/tex]. Multiply by the inverse of 4 mod 5 (4), then by 2.[tex]x=2\cdot7\cdot4\cdot2+5\cdot6[/tex]
Taken mod 7, the first term vanishes and [tex]30\equiv2\pmod7[/tex]. Multiply by the inverse of 2 mod 7 (4), then by 6.[tex]x=2\cdot7\cdot4\cdot2+5\cdot6\cdot4\cdot6[/tex]
[tex]\implies x\equiv832\pmod{5\cdot7}\implies\boxed{x\equiv27\pmod{35}}[/tex]
b.
[tex]x^5\equiv3\pmod{64}[/tex]
We have [tex]\varphi(64)=32[/tex], so by Euler's theorem,
[tex]x^{32}\equiv1\pmod{64}[/tex]
Now, raising both sides of the original congruence to the power of 6 gives
[tex]x^{30}\equiv3^6\equiv729\equiv25\pmod{64}[/tex]
Then multiplying both sides by [tex]x^2[/tex] gives
[tex]x^{32}\equiv25x^2\equiv1\pmod{64}[/tex]
so that [tex]x^2[/tex] is the inverse of 25 mod 64. To find this inverse, solve for [tex]y[/tex] in [tex]25y\equiv1\pmod{64}[/tex]. Using the Euclidean algorithm, we have
64 = 2*25 + 14
25 = 1*14 + 11
14 = 1*11 + 3
11 = 3*3 + 2
3 = 1*2 + 1
=> 1 = 9*64 - 23*25
so that [tex](-23)\cdot25\equiv1\pmod{64}\implies y=25^{-1}\equiv-23\equiv41\pmod{64}[/tex].
So we know
[tex]25x^2\equiv1\pmod{64}\implies x^2\equiv41\pmod{64}[/tex]
Squaring both sides of this gives
[tex]x^4\equiv1681\equiv17\pmod{64}[/tex]
and multiplying both sides by [tex]x[/tex] tells us
[tex]x^5\equiv17x\equiv3\pmod{64}[/tex]
Use the Euclidean algorithm to solve for [tex]x[/tex].
64 = 3*17 + 13
17 = 1*13 + 4
13 = 3*4 + 1
=> 1 = 4*64 - 15*17
so that [tex](-15)\cdot17\equiv1\pmod{64}\implies17^{-1}\equiv-15\equiv49\pmod{64}[/tex], and so [tex]x\equiv147\pmod{64}\implies\boxed{x\equiv19\pmod{64}}[/tex]
The functions q and r are defined as follows.
q(x) = -2x +1
r(x) = 2x^2 - 1
Find the value of .
q(r(4))
Answer:
q(r(4)) = -61
Step-by-step explanation:
q(x) = -2x +1
r(x) = 2x^2 - 1
q(r(4))
First find r(4)
f(4) = 2 (4)^2 -1
= 2 *16 -1
= 32-1
= 31
Then put this value in for x in q(x)
q(r(4)) = q(31) = -2(31)+1
= -62+1
= -61
Answer:
The value of q( r(4) ) = -61
Step-by-step explanation:
It is given that,
q(x) = - 2x +1
r(x) = 2x^2 - 1
To find the value of q(r(4))
r(x) = 2x^2 - 1
r(4) = 2( 4^2) - 1 [Substitute 4 instead of x]
= 2(16) - 1
= 32 - 1 = 31
q( x ) = -2x +1
q( r(4) ) = q(31) [Substitute 31 instead of x)
= (-2*31) +1
= -62 + 1 = -61
Therefore the value of q(r(4)) = -61
Please help me with this
Answer:
The correct answer is first option
24
Step-by-step explanation:
From the figure we get, mAXM = 72° and m<AMR = 38°
Also it is given that, all triangles are isosceles triangles and
m<FXA = 96°
To find the measure of <FXM
From the figure we get,
m<FXA = m<AXM + m<FXM
m<FXM = m<FXA - m<AXM
= 96 - 72
= 24
Therefore the correct answer is first option
24
A net force of 24 N is acting on a 4.0-kg object. Find the acceleration in m/s 6m/s^2
Step-by-step explanation:
mass of the object=m=4 kg
the net force exerted on it=F=24N=kgm/s²
the acceleration of the object=a=?
formula for calculating the acceleration=
F=ma
plug in the values
24kgm/s²=4kga
divide each side by 4
24kgm/s²/4 kg=4kga/4kg
a=6m/s²
Final answer:
The acceleration of the 4.0-kg object with a net force of 24 N is 6 m/s².
Explanation:
Acceleration: To find the acceleration, we use Newton's Second Law: F = ma. Given a net force of 24 N on a 4.0-kg object, the acceleration can be calculated as follows:
a = F/m = 24 N / 4.0 kg = 6 m/s²
Therefore, the acceleration of the object is 6 m/s².
$10000 is deposited in an account earning 4% interest compounded continuously. Use the continuous interest formula below to determine how long it takes for the amount in the account to double. Round answer to 2 decimal places. A = P e r t
_____years.
Answer:
The required number of years are 7.52 years.
Step-by-step explanation:
Given : $10000 is deposited in an account earning 4% interest compounded continuously.
To find : How long it takes for the amount in the account to double?
Solution :
Applying Continuous interest formula,
[tex]A=Pe^{rt}[/tex]
Where, P is the principal P=$10000
r is the interest rate r=4%=0.04
t is the time
We have given, Amount in the account to double
i.e. A=2P
Substitute the value in the formula,
[tex]2P=Pe^{rt}[/tex]
[tex]2=e^{0.04t}[/tex]
Taking log both side,
[tex]\log 2=\log (e^{0.04t})[/tex]
[tex]\log 2=0.04t\times log e[/tex]
[tex]t=\frac{\log 2}{0.04}[/tex]
[tex]t=7.52[/tex]
Therefore, The required number of years are 7.52 years.
Giovanni and Jean started a 120-mile bicycle race at the same time. Giovanni cycled at 23.9 miles per hour while Jean cycled at 24 miles per hour. When Jean crossed the finish line, how many miles was Giovanni from the finish line? Express your answer as a decimal to the nearest tenth.
Answer:
Giovanni was 0.5 miles from the finish line
Step-by-step explanation:
This is a problem of movement with constant velocity.
For this kind of problems, generally it is enough to remember the definition of average velocity v:
[tex]v=\frac{x}{t}[/tex]
Where x is the change in position that took place in an interval t.
First, find the time that Jean, who cycled at 24 miles per hour, spent on the race:
Isolating t from the last equation,
[tex]t=\frac{x}{v}[/tex], and replacing the data for Jean movement:
[tex]t=\frac{120}{24}=5h[/tex]
Second, find what was the distance that Giovanni had cycled when Jean crossed the line:
[tex]x=v*t\\ x=23.9*5=119.5[/tex]
When Jean crossed the line he had cycled 120 miles, and Giovanni 119.5; so Giovanni was 0.5 miles from the finish line.
10 men and 12 women will be seated in a row of 22 chairs. What is the probability that all men will be seated side by side in 10 consecutive positions? 1/C(22, 10) 10!/C(22, 10) 10!/22! 10! middot 12!/22! 10! middot 13!/22!
Answer:
The correct option is 4.
Step-by-step explanation:
It is given that 10 men and 12 women will be seated in a row of 22 chairs.
Total possible ways to arrange n terms is n!.
Similarly,
Total possible ways to place 22 people on 22 chairs = 22!
[tex]\text{Total outcomes}=22![/tex]
It is given that all men will be seated side by side in 10 consecutive positions.
Total possible ways to place 10 people on 10 chairs = 10!
Let 10 men = 1 unit because all men will be seated side by side in 10 consecutive positions. 12 women = 12 units because women can any where.
Total number of units = 12 + 1 = 13.
Total possible ways to place 13 units = 13!
Total possible ways to place 10 men and 12 women, when all men will be seated side by side in 10 consecutive positions is
[tex]\text{Favorable outcomes}=10!\cdot 13![/tex]
The probability that all men will be seated side by side in 10 consecutive positions
[tex]P=\frac{\text{Favorable outcomes}}{\text{Total outcomes}}=\frac{10!\cdot 13!}{22!}[/tex]
Therefore the correct option is 4.
A survey among freshman at a certain university revealed that the number of hours spent studying before final exams was normally distributed with mean 25 and standard deviation 15. A sample of 36 students was selected. What is the probabiliy that the average time spent stydying for the sampe was between 28.2 and 30 hours
Answer: 0.0775
Step-by-step explanation:
Given : Mean : [tex]\mu = 25[/tex]
Standard deviation : [tex]\sigma =15[/tex]
Sample size : [tex]n=36[/tex]
Since its normal distribution , then the formula to calculate the z-score is given by :-
[tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]
For x= 28.2 hours
[tex]z=\dfrac{28.2-25}{\dfrac{15}{\sqrt{36}}}=1.28[/tex]
For x= 30 hours
[tex]z=\dfrac{30-25}{\dfrac{15}{\sqrt{36}}}=2[/tex]
The P- value = [tex]P(1.28<z<2)[/tex]
[tex]=P(z<2)-P(z<1.28)= 0.9772498-0.8997274=0.0775224\approx0.0775[/tex]
Hence, the probabiliy that the average time spent stydying for the sampe was between 28.2 and 30 hours = 0.0775
The probability that the average time spent studying for the sample was between 28.2 and 30 hours is calculated as 0.0775 or 75 %.
To calculate the probability that the average time spent studying for the sample was between 28.2 and 30 hours, we use the normal distribution and standardize the sample means to a z-score.
Given the population mean (μ) is 25 hours, the population standard deviation (σ) is 15 hours, and the sample size (n) is 36, the standard error of the mean (SEM) is σ/√n which is 15/6 = 2.5 hours.
The z-scores for 28.2 and 30 hours are calculated as (X - μ)/(SEM).
Z for 28.2 hours = (28.2 - 25)/2.5 = 1.28
Z for 30 hours = (30 - 25)/2.5 = 2
Now we can look up these z-scores in the standard normal distribution table (or use calculator/software) to find the probabilities for these z-scores and then find the probability that lies between them by subtracting the two.
Example: Let's assume the probability corresponding to z=1.28 is 0.8997 and to z=2 is 0.9772.
The probability that the sample mean lies between 28.2 and 30 hours is:
P(1.28 < Z < 2) = P(Z < 2) - P(Z < 1.28)
= 0.9772 - 0.8997
= 0.0775
Hence, there is a 7.75% probability that the sample mean is between 28.2 and 30 hours.
For the month of November in a certain city, 53% of the days are cloudy. Also in the month of November in the same city, 46% of the days are cloudy and snowy. What is the probability that a randomly selected day in November will be snowy if it is cloudy?
Answer: The required probability that a randomly selected day in November will be snowy if it is cloudy is 86.79%.
Step-by-step explanation: Given that for the month of November in a certain city, 53% of the days are cloudy. Also in the month of November in the same city, 46% of the days are cloudy and snowy.
We are to find the probability that a randomly selected day in November will be snowy if it is cloudy.
Let A denote the event that the day is cloudy and B denote the event that the day is snowy.
Then, according to the given information, we have
[tex]P(A)=53\%=0.53,\\\\P(A\cap B)=46\%=0.46.[/tex]
Now, we need to find the conditional probability of event B given that the event A has already happened.
That is, P(B/A).
We know that
[tex]P(B/A)=\dfrac{P(B\cap A)}{P(A)}=\dfrac{0.46}{0.53}=0.87=87.79\%.[/tex]
Thus, the required probability that a randomly selected day in November will be snowy if it is cloudy is 87.79%.
Final answer:
The probability that a randomly selected cloudy day in November will be snowy is calculated using conditional probability. The result is approximately 86.79%.
Explanation:
To determine the probability that a randomly selected day in November will be snowy if it is cloudy, we use the given information: 53% of days are cloudy and 46% of days are both cloudy and snowy. The probability we are looking for is the conditional probability of it being snowy given that it is cloudy, which can be calculated by dividing the probability of it being both cloudy and snowy by the probability of it being cloudy, which is P(Snowy | Cloudy) = P(Cloudy and Snowy) / P(Cloudy).
So the calculation would be:
P(Snowy | Cloudy) = (0.46) / (0.53)
= 0.8679 (or 86.79%).
Therefore, there is an 86.79% chance that it will be snowy on a day that is cloudy in that city in November.
Lauren Industries wants to open a warehouse in the U.S. northwest. The location factors, weights, and potential locations are shown below.
Location Factor
Weight
Olympia
Salem
Boise
Helena
Land Availability/Cost
0.25
80
100
90
87
Area Population
0.20
80
85
100
82
Quality of Life
0.15
100
88
92
100
Labor Availability
0.25
91
100
80
90
Proximity to Major Highway
0.15
85
88
100
91
Calculate the total weighted-factor score for Salem.
86.50
93.40
91.30
89.30
100
Answer:
Find the slope of the line that passes through the points shown in the table.
The slope of the line that passes through the points in the table is
.
Step-by-step explanation:
In a class of 40 students, everyone has either a pierced nose or a pierced ear. The professor asks everyone with a pierced nose to raise his or her hand. Eight hands go up. Then the professor asked everyone with a pierced ear to do likewise. This time there are 35 hands raised. How many students have piercings both on their ears and their noses?
Answer:
3 students
Step-by-step explanation:
If everyone in the class has either a pierced nose or ear, we just simply have to add up the total number of hands raised and minus the number of students in the class.
35+8=43
43-40=3
3 students have both a pierced nose and pierced ear.
Find a power series representation for the function. (Give your power series representation centered at x = 0.)f(x) = x3x2 + 1f(x) = ∞n = 0 Determine the interval of convergence. (Enter your answer using interval notation.)
I suppose you mean
[tex]f(x)=\dfrac{x^3}{x^2+1}[/tex]
Recall that for [tex]|x|<1[/tex], we have
[tex]\dfrac1{1-x}=\displaystyle\sum_{n=0}^\infty x^n[/tex]
Then
[tex]\dfrac1{1+x^2}=\dfrac1{1-(-x^2)}=\displaystyle\sum_{n=0}^\infty(-x^2)^n=\sum_{n=0}^\infty(-1)^nx^{2n}[/tex]
which is valid for [tex]|-x^2|=|x|^2<1[/tex], or more simply [tex]|x|<1[/tex].
Finally,
[tex]f(x)=\displaystyle\frac{x^3}{x^2+1}=\sum_{n=0}^\infty(-1)^nx^{2n+3}[/tex]
Consider the integral 8 (x2+1) dx 0 (a) Estimate the area under the curve using a left-hand sum with n = 4. 250 Is this sum an overestimate or an underestimate of the true value? overestimate underestimate (b) Estimate the area under the curve using a right-hand sum with n = 4. 248
Answer:
(a) 120 square units (underestimate)
(b) 248 square units
Step-by-step explanation:
(a) left sum
See the attachment for a diagram of the areas being summed (in orange). This is the sum of the first 4 table values for f(x), each multiplied by 2 (the width of the rectangle). Quite clearly, the curve is above the rectangle for the entire interval, so the rectangle area underestimates the area under the curve.
left sum = 2(1 + 5 + 17 + 37) = 2(60) = 120 . . . . square units
(b) right sum
The right sum is the sum of the last 4 table values for f(x), each multiplied by 2 (the width of the rectangle). This sum is ...
right sum = 2(5 +17 + 37 +65) = 2(124) = 248 . . . . square units
In order to start a small business, a student takes out a simple interest loan for $3000.00 for 9 months at a rate of 11.75%. a. How much interest must the student pay? b. Find the future value of the loan. a. The amount of interest is $ . (Round to the nearest cent as needed.) b. The future value is $ (Round to the nearest cent as needed.)
Answer: (a) $264.375 ⇒ Amount of Interest
(b) Future Value = $3264.375
Step-by-step explanation:
(a) Principal amount = $3000
Time period = 9 months
Interest rate = 11.75%
Simple interest(SI) = principal amount × rate of interest (i) × time period
= 3000 × [tex]\frac{11.75}{100}[/tex] × [tex]\frac{9}{12}[/tex]
= 3000 × 0.1175 × 0.75
= $264.375 ⇒ Amount of Interest
(b) Future value of loan = principal amount + interest amount
= 3000 + 264.375
= 3264.375
A floor refinishing company charges $1.83 per square foot to strip and refinish a tile floor for up to 1000 square feet. There is an additional charge of $350 for toxic waste disposal for any job which includes more than 150 square feet of tile.
A) Express the cost, y, of refinishing a floor as a function of the number of square feet, x, to be refinished.
b) Graph the function, give the domain and range.
Answer:
Here x represents the number of square feet to be refinished and y represents the cost of refinishing the floor,
Given,
The cost of a tile floor for up to 1000 square feet is $1.83 per square,
So, the cost of x square feet of tile = 1.83x for x ≤ 1000
⇒ y = 1.83x for x ≤ 1000
Since, there is an additional charge of $350 for toxic waste disposal for any job which includes more than 150 square feet of tile.
That is, y = 1.83x + 350, for x > 150
So, y must be 1.83x for x ≤ 150.
A) Hence, the function that express the cost, y, of refinishing a floor as a function of the number of square feet, x, to be refinished, is,
[tex]y=\begin{cases}1.83x & \text{ if } 0\leq x\leq 150 \\ 1.83x+350 & \text{ if } 150< x\leq 1000\end{cases}-----(1)[/tex]
B) The domain of the function = all possible value of x
⇒ Domain = 0 ≤ x ≤ 1000
Range = All possible value of y,
Since, the range of function y=1.83x, 0≤ x ≤ 150 is [0, 274.5]
While the range of function y = 1.83x + 350, for x > 150 is (624.5, 2180]
Hence, the range of the function (1) = [0, 274.5]∪(624.5, 2180]
The cost of refinishing a floor can be expressed as a piecewise function based on the number of square feet to be refinished. The domain of the function is all real numbers, and the range is all real numbers greater than or equal to 0.
Explanation:Let x represent the number of square feet to be refinished.
For x ≤ 150, the cost of refinishing a floor is simply $1.83 per square foot. So, the cost function, y, for x ≤ 150 is y = 1.83x.
For x > 150, there is an additional charge of $350 for toxic waste disposal. So, the cost function, y, for x > 150 is y = 350 + 1.83x.
The overall cost function, y, is given by:
y = 1.83x, for x ≤ 150
y = 350 + 1.83x, for x > 150
The domain of the function is all real numbers, since any positive number of square feet can be refinished. The range of the function is all real numbers greater than or equal to 0, since the cost cannot be negative.
The following data summarizes results from 941 pedestrian deaths that were caused by accidents. If one of the pedestrian deaths is randomly selected, find the probability that the pedestrian was intoxicated or the driver was intoxicated.
Pedestrian Pedestrian
Intoxicated Not intoxicated
Driver Intoxicated 56 71
Driver Not intoxicated 292 522
Answer:
P=0.3698 or 36.98%
Step-by-step explanation:
Complete the table by adding the totals to each column and row.
Pedestrian Pedestrian
Intoxicated Not intoxicated Totals
Driver Intoxicated 56 71 127
Driver Not intoxicated 292 522 814
Totals 348 593 941
The probability that the pedestrian was intoxicated or the driver was intoxicated is the opposite event of neither of them was intoxicated. The total of cases when neither of them was intoxicated is 593. So the probability is:
P1=593/941=0.6302
The probability of the opposite event is one minus the probability calculated:
P=1-0.6302=0.3698
And this is the probability that the pedestrian was intoxicated or the driver was intoxicated.
Solve the following inequalities:
A) 2w + 17 ˃ -4w -25
B) 2.3 + 0.6t ˂ 2 + 0.8t
C) Determine if – 3.5 and 10 make the inequalities in Parts A) and B) true or not. Check both numbers in both inequalities. Show your work.
Answer:
A) w > -7
B) 1.5 < t
C) -3.5 makes A) true and B) false
10 makes both inequalities true
Step-by-step explanation:
The idea of these exercises is to clear our variable, we need it to be alone on one side of the inequality
A) 2w + 17 ˃ -4w -25
First, we will put together on one side the terms with a w and on the other the terms without w.
For that, we have to add 4w - 17 on both sides
2w + 17 + 4w - 17 ˃ -4w -25 + 4w - 17 (Notice that 17-17=0 and -4w+4w=0, so we don't have to write them below)
2w + 4w > -25 - 17
Now we can sum the terms (we didn't do it before because we can't sum a term with a w with one without it)
6w > -42
We divide by 6 on both sides and we have
6/6w > -42/6
w > -7
B) 2.3 + 0.6t ˂ 2 + 0.8t
We start as before; in this case we have to put together the terms with a t (our variable changes name but the idea is the same)
We will add -2 - 0.6t on both sides
2.3 + 0.6t -2 - 0.6t ˂ 2 + 0.8t -2 - 0.6t
2.3 - 2 < 0.8t - 0.6t
Now we sum the terms
0.3 < 0.2t
We divide by 0.2 on both sides and we have
0.3/0.2 < 0.2/0.2t
1.5 < t
C) Let's check -3.5 on both inequalities:
We have to replace the variable by -3.5:
2*(-3.5) + 17 ˃ -4*(-3.5) -25 (remember that if there is no sign between a number and a variable, it means that is a multiplication)
Now we just solve the calculation
-7 + 17 > 14 -25
10 > -11
That's true, so -3.5 makes the inequality true.
Now, in the other inequality, we replace the t by -3.5 and solve as before
2.3 + 0.6*(-3.5) ˂ 2 + 0.8*(-3.5)
2.3 - 2.1 < 2 - 2.8
0.2 < -0.8
That's false because we are saying that a negative number is bigger than a positive one, so -3.5 makes the inequality not true.
Now we do the same with 10 in both inequalities:
2*10 + 17 ˃ -4*10 -25
20 + 17 > -40 -25
37 > - 65
It's true!
2.3 + 0.6*10 ˂ 2 + 0.8*10
2.3 + 6 < 2 + 8
8.3 < 10
It's true!
Q2. On a cold day, hailstones fall with a velocity of (2i− 6k) m s−1 . If a cyclist travels through the hail at 10i ms−1 , what is the velocity of the hail relative to the cyclist? At what angle are the hailstones falling relative to the cyclist
Answer:[tex]-8\hat{i}-6\hat{k}[/tex]
[tex]\theta =\tan^{-1}\left ( \frac{3}{4} \right )[/tex]
Step-by-step explanation:
Given
Velocity of hailstones fall[tex]\left ( V_h\right )=2\hat{i}-6\hat{k}[/tex] m/s
Velocity of cyclist [tex]\left ( V_c\right )=10\hat{i}[/tex] m/s
Therefore
Velocity of hail with respect to cyclist[tex]\left ( V_{hc}\right )[/tex]
[tex]V_{hc}=V_h-V_c[/tex]
[tex]V_{hc}=2\hat{i}-6\hat{k}-10\hat{i}[/tex]
[tex]V_{hc}=-8\hat{i}-6\hat{k}[/tex]
and angle of hails falling relative to the cyclist is given by
[tex]\theta =\tan^{-1}\left ( \frac{3}{4}\right )[/tex]
[tex]\theta [/tex] is the angle made with the vertical
Ted is not particularly creative. He uses the pickup line "If I could rearrange the alphabet, I'd put U and I together." The random variable x is the number of girls Ted approaches before encountering one who reacts positively. Determine whether the table describes a probability distribution. If it does, find its mean and standard deviation. If a probability distribution is not given, identify the requirements that are not satisfied.
x P(x)
1 0.001
2 0.025
3 0.101
4 0.246
5 0.503
Answer:
Not a probability distribution
Step-by-step explanation:
The given table doesn't describe a probability distribution as in order for the given distribution to be a probability distribution the sum of probabilities is required to be equal to one.
Here,
Sum of probabilities = 0.001+0.025+0.101+0.246+0.503 = 0.876
The sum of probabilities is not equal to one.
Therefore, the given distribution is not a probability distribution ..
Individuals who have a certain gene have a 0.46 probability of contracting a certain disease. Suppose that 913 individuals with the gene participate in a lifetime study. What is the standard deviation of the number of people who eventually contract the disea
Answer: 15.06
Step-by-step explanation:
Given : The number of individuals participated in a lifetime study :[tex]n=913[/tex]
The probability of individuals with certain gene of contracting a certain disease :[tex]p= 0.46[/tex]
Now, the standard deviation of the number of people who eventually contract the disease is given by :_
[tex]\sigma =\sqrt{np(1-p)}\\\\=\sqrt{913\times0.46(1-0.46)}=15.059521904\approx15.06[/tex]
Hence, the the standard deviation of the number of people who eventually contract the disease = 15.06
Jack typed 80 words per minute when he enrolled in a typing course. His typing speed increased by 3% two weeks into the course. At the end of the course, Jack was able to type his entire 1, 680 word document In 20 minutes. What was the percent of increase in his typing speed from the beginning of the course to the end? a 3% b 4% c 5% d 6% e 7%
Answer:
The answer is option C : 5%.
Step-by-step explanation:
Jack typed 80 words per minute when he enrolled in a typing course.
His typing speed increased by 3% two weeks into the course.
At the end of the course, Jack was able to type his entire 1, 680 word document In 20 minutes.
Hence, the percent of increase in his typing speed from the beginning of the course to the end is given by:
[tex]1680/20=84[/tex]
[tex](84-80)/80[/tex] =5%
Therefore, the answer is option C : 5%.
To find the percent increase in Jack's typing speed, compare his initial and final speeds. The percent increase is 5%.
Explanation:To find the percent increase in Jack's typing speed, we need to compare his initial speed to his final speed. Let's start by calculating his initial typing speed:
80 words per minute
To find his final typing speed, we need to determine how many words he typed in 20 minutes:
1,680 words / 20 minutes = 84 words per minute
Now we can find the percent increase:
(Final speed - Initial speed) / Initial speed * 100
(84 - 80) / 80 * 100 = 5%
Therefore, the percent of increase in Jack's typing speed from the beginning of the course to the end is 5%.
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A bag contains 6 red apples and 5 yellow apples. 3 apples are selected at random. Find the probability of selecting 1 red apple and 2 yellow apples.
To solve the problem, first calculate all possible combinations of selecting 3 apples from 11. Then calculate the favorable combinations, which include selecting 1 red apple (from 6 available) and 2 yellow apples (from 5 available). Divide these values to get the probability.
Explanation:The topic at hand is one of probability, more specifically, it's a problem of combinations in probability. The bag contains a total of 11 apples (6 red and 5 yellow). When 3 apples are chosen, we want to find the probability that 1 is red and 2 are yellow.
First, calculate the total number of ways to choose 3 apples from 11, which is denoted as '11 choose 3', using combination formula C(n,r) = n! / [r!(n - r)!]. Then, consider the number of favorable outcomes: choosing 1 red apple from 6 (denoted as '6 choose 1') and 2 yellow apples from 5 (denoted as '5 choose 2'). Multiply these two results because we choose '1 red' and '2 yellow', using the rule of product. Calculate these individual results and then divide the favorable outcomes by the total outcomes to get the required probability.
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A thief steals an ATM card and must randomly guess the correct seven-digit pin code from a 4-key keypad. Repetition of digits is allowed. What is the probability of a correct guess on the first try?
The probability of guessing the correct seven-digit pin code on the first try is very low, approximately 0.000024%.
Given that,
A thief steals an ATM card.
The thief must guess the correct seven-digit pin code.
The pin code is entered using a 4-key keypad.
The probability of guessing the correct seven-digit pin code on the first try depends on a few factors.
To break it down,
if the thief has a 4-key keypad and repetition of digits is allowed, that means there are four options for each digit.
So, there are a total of 4⁷ (4 raised to the power of 7) possible combinations.
Since the thief is trying to guess the correct pin code on the first try, there is only one correct combination out of the total possible combinations.
Therefore,
The probability of guessing the correct pin code on the first try would be 1 out of 4⁷, or approximately 0.00000024, or 0.000024%.
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The probability of randomly guessing a 7-digit PIN from a 4-key keypad is 1 in 16,384. This equals approximately 0.000061 or 0.0061%. Each digit has 4 possible options, and there are 7 digits in total.
The probability of guessing a seven-digit PIN code correctly from a 4-key keypad (where repetition of digits is allowed) can be calculated as follows:
Since each of the 7 digits in the PIN can be any of 4 possible digits (0 through 3), the total number of possible combinations is calculated by raising the number of choices per digit to the power of the number of digits:
→ Total possible combinations = 4^7
= 4⁷
= 16384
Therefore, the probability of guessing the correct PIN on the first try is the reciprocal of the total number of possible combinations:
→ Probability of a correct guess = 1 / 16384
Hence, the probability is approximately 0.000061 or 0.0061%.
Add the Base 2 numbers: 11,110,110 +101,101,111
Answer:
100000
Step-by-step explanation:
I did the operations in the picture, you only have to know that 0+1=01, 1+1=10
and 1+1+1=11.
Now, I don't know if you need to calculate the total add, I calculated it.
In this case, you need to know that 1+1+1+1=100.
The sum of the two Base 2 numbers is: [tex]\[ {10001001_2} \][/tex].
The sum of the given Base 2 (binary) numbers is calculated as follows:
11,110,110
+ 101,101,111
Starting from the rightmost digit (least significant bit) and moving left, we add the digits:
- In the rightmost column, 0 + 1 = 1.
- In the next column, 1 + 1 = 10 (which is 0 in the current column and carry over 1 to the next column).
- Continuing this process, we add the digits along with any carry from the previous column.
Let's continue the addition:
11,110,110
+ 101,101,111
--------------
1,000,1001
Here's the step-by-step process:
- 0 + 1 = 1 (no carry).
- 1 + 1 = 10 (0 in this column, carry 1).
- 1 + 0 + 1 (carry) = 10 (0 in this column, carry 1).
- 1 + 1 + 1 (carry) = 11 (1 in this column, carry 1).
- 1 + 0 + 1 (carry) = 10 (0 in this column, carry 1).
- 1 + 1 + 1 (carry) = 11 (1 in this column, carry 1).
- 1 + 0 + 1 (carry) = 10 (0 in this column, carry 1).
- 1 + 1 + 1 (carry) = 11 (1 in this column, carry 1).
- 1 + 0 + 1 (carry) = 10 (0 in this column, carry 1).
- 1 + 1 + 1 (carry) = 11 (1 in this column, carry 1).
- Finally, we have a carry of 1 that we add to the leftmost digit, giving us 1 + 1 = 10 (0 in this column, carry 1).
Since there are no more digits to add, we write down the 1 at the beginning:
1,000,1001
Therefore, the sum of the two Base 2 numbers is:
[tex]\[ {10001001_2} \][/tex].
Problem 4. Solve the following counting problems. (a) How many committees of size 6 can be formed by a club consisting of 15 people?
Answer:
5005
Step-by-step explanation:
In the question we have to form a committee of six from a club of consisting of 15 people.
This a simple case of selection of six people from a group of 15 people.
which can be done in
[tex]^{15}C_6= \frac{15!}{6!\times9!}=5005[/tex]
hence, the number of ways of forming committees of size six from a club of 15 members= 5005
A quality control inspector has drawn a sample of 10 light bulbs from a recent production lot. Suppose 20% of the bulbs in the lot are defective. What is the probability that exactly 7 bulbs from the sample are defective? Round your answer to four decimal places
Answer:
The probability is 0.0008.
Step-by-step explanation:
Let X represents the event of defective bulb,
Given, the probability of defective bulb, p = 20 % = 0.2,
So, the probability that bulb is not defective, q = 1 - p = 0.8,
The number of bulbs drawn, n = 10,
Since, binomial distribution formula,
[tex]P(x=r) = ^nC_r p^r q^{n-r}[/tex]
Where, [tex]^nC_r = \frac{n!}{r!(n-r)!}[/tex]
Hence, the probability that exactly 7 bulbs from the sample are defective is,
[tex]P(X=7)=^{10}C_7 (0.2)^7 (0.8)^{10-7}[/tex]
[tex]=120 (0.2)^7 (0.8)^3[/tex]
[tex]=0.000786432[/tex]
[tex]\approx 0.0008[/tex]
Calculate the probability of exactly 7 defective bulbs in a sample of 10 using the binomial distribution formula.
Binomial distribution:
Calculate the probability that exactly 7 out of 10 bulbs are defective using the binomial distribution formula.Use the formula: P(X = k) = C(n, k) * p^k * (1-p)^(n-k) where n = 10, k = 7, and p = 0.2.Calculate the probability:P(X = 7) = C(10, 7) * (0.2)^7 * (0.8)^3 ≈ 0.2013
(×-5) second power equals 3
Answer: [tex]x_1=6.73\\\\x_2=3.26[/tex]
Step-by-step explanation:
Given the following expression:
[tex](x-5)^2=3[/tex]
And knowing that:
[tex](a\±b)^2=a^2-2ab+b^2[/tex]
We get:
[tex]x^2-2(x)(5)+5^2=3\\\\x^2-10x+25=3[/tex]
Move the 3 to the left side of the equation:
[tex]x^2-10x+25-3=0\\\\x^2-10x+22=0[/tex]
Apply the Quadratic formula:
[tex]x=\frac{-b\±\sqrt{b^2-4ac} }{2a}[/tex]
In this case:
[tex]a=1\\b=-10\\c=22[/tex]
Substituting values into the Quadratic formula, we get:
[tex]x=\frac{-(-10)\±\sqrt{(-10)^2-4(1)(22)} }{2(1)}\\\\\\x_1=6.73\\\\x_2=3.26[/tex]