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The probability that a woman has none of the three risk factors, given that she does not have risk factor A, is calculated to be 0.54.
Explanation:To find the probability that a woman has none of the three risk factors, given that she does not have risk factor A, we will use the given probabilities and apply the principles of probability. Let's denote the probabilities of having only the risk factors as P(A), P(B), and P(C), the probabilities of having exactly two risk factors as P(A and B), P(A and C), and P(B and C), and the probability of having all three risk factors as P(A and B and C).
Given:
P(A) = P(B) = P(C) = 0.1
P(A and B) = P(A and C) = P(B and C) = 0.12
P(A and B and C | A and B) = 1/3
We can calculate P(A and B and C) using the conditional probability:
P(A and B and C) = P(A and B) × P(A and B and C | A and B) = 0.12 × 1/3 = 0.04
To find the probability of not having A, denoted as P(A'), we can use the complement rule:
P(A') = 1 - P(A) - P(A and B) - P(A and C) - P(A and B and C) = 1 - 0.1 - 0.12 - 0.12 - 0.04 = 0.62
Since P(A') includes probabilities of women with neither of the risk factors or only with B or C, we need to subtract the probabilities of having only risk factors B and C:
P(None | A') = P(A') - P(B) - P(C) + P(B and C) = 0.62 - 0.1 - 0.1 + 0.12 = 0.54
The probability that a woman has none of the three risk factors, given that she does not have risk factor A, is thus 0.54.
The angle of inclination from the base of skyscraper A to the top of skyscraper B is approximately 10.4degrees. If skyscraper B is 1472 feet tall, how far apart are the two skyscrapers? Assume the bases of the two buildings are at the same elevation.
Answer:
8020 feet
Step-by-step explanation:
The tangent relation can be used to answer this question, since it relates the sides of a right triangle to the acute angle.
tan(elevation angle) = (1472 ft)/(distance between)
Then ...
distance between = (1472 ft)/tan(10.4°) ≈ 8020 ft
The skyscrapers are 8020 feet apart.
Using the tangent of the given angle of inclination and the height of skyscraper B, the horizontal distance between the bases of the two skyscrapers is calculated to be approximately 8077 feet.
Explanation:The question is looking for the horizontal distance between the bases of two skyscrapers, given the height of one skyscraper and the angle of inclination from its base to the top of the other. This scenario forms a right triangle, where the height of skyscraper B is the opposite side, the distance between the skyscrapers is the adjacent side, and the angle of inclination is the given angle. We can use the tangent trigonometric function, which is the ratio of the opposite side to the adjacent side, to solve for the distance.
To calculate the distance (adjacent side) we can rearrange the equation: Tan(angle) = opposite/adjacent, to: Adjacent = opposite/tan(angle). Plugging in our given values we find: Distance = 1472 feet / tan(10.4 degrees) = approximately 8077 feet. Thus, the two skyscrapers are about 8077 feet apart.
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A girl flies a kite at a height 34 m above her hand. If the kite flies horizontally away from the girl at the rate of 3 m/s, at what rate is the string being let out when the length of the string released is 60 m? Assume that the string remains taut.
Answer:
2.47 m/s
Step-by-step explanation:
A girl flies a kite at a height 34 m above her hand.
It is vertical height of kite, 34 m
The horizontal rate of kite, [tex]\dfrac{dx}{dt}=3/ m/s[/tex]
Let the length of string released be s m
In right triangle using pythagoreous theorem
[tex]s^2=34^2+x^2[/tex]
For s = 60 m ,
[tex]60^2=34^2+x^2[/tex]
[tex]x=49.44[/tex] m
Differentiate the equation [tex]s^2=34^2+x^2[/tex] w.r.t t
[tex]2s\dfrac{ds}{dt}=0+2x\dfrac{dx}{dt}[/tex]
[tex]2\cdot 60\cdot \dfrac{ds}{dt}=2\cdot 49.44\cdot 3[/tex]
[tex]\dfrac{ds}{dt}=\dfrac{296.62}{120}[/tex]
[tex]\dfrac{ds}{dt}=2.47[/tex] m/s
Hence, the rate of string letting out 2.47 m/s
Betty measured the diagonal length of a playing card to be 6 inches. The short side of the card is 4 inches. What is the length of the side of the playing card?
Answer:
The length of the longer side is 4.48 inches.
Step-by-step explanation:
Given,
Length of diagonal = 6 in
Length of Short side = 4 in
Solution,
Let the length of long side be x.
Since the card is in the shape of rectangle. On drawing the diagonal the rectangle divides into two equal triangle.
So for find out the length of other side we use the Pythagoras theorem, which states that;
"In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides."
[tex]Hypotenuse^2=(Short\ side)^2+(Long\ side)^2[/tex]
[tex]\therefore 6^2=4^2+x^2\\36=16+x^2\\x^2=36-16=20\\x=\sqrt{20} =2\sqrt{5}[/tex]
[tex]x=2\times2.24=4.48\ in[/tex]
Thus the length of the longer side is 4.48 inches.
The number of years a radio functions is exponentially distributed with parameter λ = 1 8 . If Jones buys a used radio, what is the probability that it will be working after an additional 8 years?
Answer:
[tex]P(X>8)=e^{-1}[/tex]
Step-by-step explanation:
The exponential distribution is "the probability distribution of the time between events in a Poisson process (a process in which events occur continuously and independently at a constant average rate). It is a particular case of the gamma distribution". The probability density function is given by:
[tex]P(X=x)=\lambda e^{-\lambda x}, x>0[/tex]
And 0 for other case. Let X the random variable that represent "The number of years a radio functions" and we know that the distribution is given by:
[tex]X \sim Exp(\lambda=\frac{1}{8})[/tex]
We can assume that the random variable t represent the number of years that the radio is already here. So the interest is find this probability:
[tex]P(X>8|X>t)[/tex]
We have an important property on the exponential distribution called "Memoryless" property and says this:
[tex]P(X>a+t| X>t)=P(X>a)[/tex]
Where a represent a shift and t the time of interest.
On this case then [tex]P(X>8|X>t)=P(X>8+t|X>t)=P(X>8)[/tex]
We can use the definition of the density function and find this probability:
[tex]P(X>8)=\int_{8}^{\infty} \frac{1}{8}e^{-\frac{1}{8}x}dx[/tex]
[tex]=\frac{1}{8} \int_{8}^{\infty} e^{-\frac{1}{8}x}dx[/tex]
[tex]=[lim_{x\to\infty} (-e^{-\frac{1}{8}x})+e^{-1}]=0+e^{-1}=e^{-1}[/tex]
The perpendicular bisector of side AB of triangle ABC intersects the extension of side AC at D. Find the measure of angle ABC if measurement of angle CBD=16 degrees and measurement of angle ACB=118 degrees
Answer:
23°
Step-by-step explanation:
Let the interior angles of ΔABC be referenced by A, B, and C. The definition of point D means that ΔDAB is an isosceles triangle, so we have the relations ...
A + B + 118 = 180 . . . . interior angles of ΔABC
A = B +16 . . . . . . . . . . base angles of ΔDAB
Using the expression for A in the second equation to substitute into the first equation, we get ...
(B+16) +B +118 = 180
2B + 134 = 180 . . . . . collect terms
2B = 46 . . . . . . . . . . . subtract 134
B = 23 . . . . . . . . . . . . divide by 2
m∠ABC = 23°
write a proportion and solve for the question.
121.32 croatian kuna is worth US$18. How much in US dollars would you get for 375 croatian kuna?
Answer:
The worth of 375 Croatian Kuna = $ 55.54
Step-by-step explanation:
Here, given:
The worth of 121.32 Croatian Kuna = $18
Now, let us assume the worth of 375 Croatian Kuna = $ m
As, both have same units in conversion at the same rate,
So, by the RATIO OF PROPORTION:
[tex]\frac{18}{121.32 } = \frac{m}{375}[/tex]
Solving for the value of m, we get:
[tex]m = \frac{18}{121.32} \times 375 = 55.64[/tex]
or, m = $55.64
Hence, the worth of 375 Croatian Kuna = $ 55.54
Find the length of the base of a triangle when one side is 2cm shorter than the base and the other side is 3cm longer than the base. The perimeter is greater than 19cm
Answer:
> 6 cm
Step-by-step explanation:
Let b represent the length of the base in cm. Then the perimeter is ...
b + (b -2) + (b +3) > 19
3b +1 > 19 . . . . . . collect terms
3b > 18 . . . . . . . . subtract 1
b > 6 . . . . . . . . . divide by 3
The length of the base is greater than 6 cm.
The length of the base of the triangle is greater than 6 cm.
Explanation:To find the length of the base of the triangle, let's assume that the base is x cm. According to the question, one side is 2 cm shorter than the base, so its length would be (x - 2) cm. The other side is 3 cm longer than the base, so its length would be (x + 3) cm. The perimeter of a triangle is the sum of all its sides, so we can set up an equation: x + (x - 2) + (x + 3) > 19. Solving this inequality will give us the value of x, which is the length of the base.
Start by simplifying the equation: 3x + 1 > 19.Subtract 1 from both sides: 3x > 18.Divide both sides by 3: x > 6.Therefore, the length of the base of the triangle is greater than 6 cm.
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Use Euler's formula to derive the identity. (Note that if a, b, c, d are real numbers, a + bi = c + di means that a = c and b = d. Simplify your answer completely.) sin(2θ) = 2 sin(θ) cos(θ) Using Euler's formula, we have ei(2θ) = + i sin(2θ). On the other hand, ei(2θ) = (eiθ)2 = + i sin(θ) 2 = (cos2(θ) − sin2(θ)) + i sin(θ) . Equating Correct: Your answer is correct. parts, we find sin(2θ) = 2 sin(θ) cos(θ).
Answer with Step-by-step explanation:
We have to prove that
[tex]sin 2\theta=2sin\theta cos\theta[/tex] by using Euler's formula
Euler's formula :[tex]e^{i\theta}=cos\theta+isin\theta[/tex]
[tex]e^{i(2\theta)}=(e^{i\theta})^2[/tex]
By using Euler's identity, we get
[tex]cos2\theta+isin2\theta=(cos\theta+isin\theta)^2[/tex]
[tex]cos2\theta+isin2\theta=(cos^2\theta-sin^2\theta+2isin\theta cos\theta)[/tex]
[tex](a+b)^2=a^2+b^2+2ab, i^2=-1[/tex]
[tex]cos2\theta+isin2\theta=cos2\theta+i(2sin\theta cos\theta)[/tex]
[tex]cos2\theta=cos^2\theta-sin^2\theta[/tex]
Comparing imaginary part on both sides
Then, we get
[tex]sin2\theta=2sin\theta cos\theta[/tex]
Hence, proved.
Michelle has 8 1/4 pounds of dry cat food for her cat smokey. She places the cat food into 3 containers to use at later date. How much cat food will be in each container
Answer:
Each container contains [tex]2.5 \ pounds \ \ OR \ \ \frac{11}{4} \ pounds \ \ OR \ \ 2 \frac{3}{4} \ pounds[/tex] of cat food.
Step-by-step explanation:
Given:
Amount of cat food = 8 1\4 pounds.
Also 8 1\4 can be rewritten as 8.25 pounds
Number of containers =3
We need to find the amount of cat food in each container.
Amount of cat food in each container can be calculated by Dividing Amount of cat food she has with number of Containers.
Amount of cat food in each container = [tex]\frac{8.25}{3}= 2.75 \ pounds[/tex]
2.75 pounds ca be rewritten as [tex]\frac{11}{4} \ pounds \ \ OR \ \ 2 \frac{3}{4} \ pounds[/tex]
Hence Each container contains [tex]2.5 \ pounds \ \ OR \ \ \frac{11}{4} \ pounds \ \ OR \ \ 2 \frac{3}{4} \ pounds[/tex] of cat food.
Dale graphed the absolute value parent function. Then, he reflected the graph over the x-axis, shifted it four units to the right and three units up. Give the new equation
Answer:
i(x) = - |x - 4| + 3Step-by-step explanation:
Refer to attached graph
Parent function:
f(x) = |x|, solid black on the graphTransformations
1. Reflection over x-axis: f(x) → -f(x)
g(x) = -|x|, dotted blue on the graph2. Horizontal shift 4 units to the right: g(x) → g(x - 4)
h(x) = -|x - 4|, dotted green on the graph3. Vertical shift 3 units up: h(x) → h(x) + 3
i(x) = - |x - 4| + 3, solid red on the graphThis is the final function
A soccer ball is kicked in the air off a 22.0 meter high hill. The equation h(t)=-5t^2+10t+22 gives the approximated height h, in meters, of the ball t seconds after it is kicked. What equation can be used to tell if the ball reaches a height of 35 meters? Does the ball reach a height of 35 meters? How can you tell?
Equation:____
Answer:____
Answer:
Equation: 5t² − 10t + 13 = 0
Answer: No
Step-by-step explanation:
h(t) = -5t² + 10t + 22
When h(t) = 35:
35 = -5t² + 10t + 22
5t² − 10t + 13 = 0
This equation must have at least one real solution if the ball is to reach a height of 35 meters. Which means the discriminant can't be negative.
b² − 4ac
(-10)² − 4(5)(13)
100 − 260
-160
The ball does not reach a height of 35 meters.
Answer:
Equation: 5t² − 10t + 13 = 0
Answer: No
In a calculus class, Jack Hartig scored 4 on a quiz for which the class mean and standard deviation were 2.9 and 2.1, respectively. Norm Alpina scored 8 on another quiz for which the class mean and standard deviation were 6.5 and 1.9, respectively. Relatively speaking, which student did better? Make use of z-scores.
Answer: Norm Alpina did better with z-score 0.79
Step-by-step explanation:
Z score formula = (raw score - mean) / standard deviation
For Jack Hartig,
score = 4; mean = 2.9; standard deviation = 2.1
Hence, Z score = (4 - 2.9) /2.1
= 1.1/2.1
= 0.52
For Norm Alpina,
score = 8; mean = 6.5; standard deviation = 1.9
Hence, Z score = (8 - 6.5) /1.9
= 1.5/1.9
= 0.79
Relatively, Norm Alpina did better for having Z score 0.79
By calculating z-scores for both students, which represent the number of standard deviations their scores are from the mean, Norm Alpina has a higher z-score and hence performed relatively better compared to Jack Hartig on their respective quizzes.
In order to determine which student did relatively better on their quizzes, we need to calculate the z-scores for each student. A z-score indicates how many standard deviations an observation is above or below the mean. The formula for a z-score is Z = (X - μ) / σ, where X is the score, μ(mu) is the mean, and σ(sigma) is the standard deviation.
For Jack Hartig:
Z = (4 - 2.9) / 2.1
= 1.1 / 2.1
= 0.524
For Norm Alpina:
Z = (8 - 6.5) / 1.9
= 1.5 / 1.9
= 0.789
Norm Alpina's z-score is higher, indicating that, relatively speaking, he performed better than Jack Hartig on the quiz based on how their scores relate to their respective class means and standard deviations.
Find the circumference. Leave your answer in terms of pi.
The circumference of circle with radius 18 inches in terms of pi is 36π inches
Solution:From the given figure, radius "r" = 18 inches
We have to find circumference of circle
circumference would be the length of the circle if it were opened up and straightened out to a line segment.
The circumference of circle is given as:
[tex]\text {circumference of circle }=2 \pi r[/tex]
Where "r" is the radius of circle
Substituting the value r = 18 inches in above formula,
[tex]\text {circumference of circle }=2 \times \pi \times 18=36 \pi[/tex]
Thus circumference of circle in terms of pi is 36π inches
Mrs.Gonzalez has 36 students in her class and only 9 of them are boys . What percent of the students in Mrs.Gonzalez class are boys? Write a proportion and show your work please
A new pair of wireless earbuds cost $125. You found earbuds online and the website is offering a 20% discount if you buy them this week. But you remember seeing the same earbuds at your local store on sale for $90. Where will you purchases earbuds? Show work to support your answer please
Question #1:
To find the percentage of boy's in the class, divide.
9 / 36 = 0.25
0.25 * 100% = 25% (boys in the class)
We can write this proportion as 9/36 since this is the same as 25%.
_________
Question #2:
We know that the ear buds in both options cost $125.
The earbuds online are on a 20% discount.
The earbuds in store are on a 90$ sale
Lets find the discount for the online pair.
20% = 0.2
125 * 0.2 = 25
125 - 25 = $100 (price after discount)
After solving we can see that the earbuds in store are a better price. So, you will purchase the ear buds in store.
_________
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Richard walked around 2 rectangular parks. One measures 450 feet by 167 feet and the other measures 234 feet by 156 feet. Part B Richard's brother walked around a different park that is 254 feet by 56 feet.
What is the question?
Answer:
The answer is in the explanation.
Step-by-step explanation:
450x2=900 167x2=334 Then you add those together and you get 1,234 then you do 234x2=468 156x2=312 then you add those two, and you get 780 then you add 1,234+780= 2014. For his brother you do 254x2=508 and 56x2=112 then you add them and you get 620. Then you subtract 2014-620= 1394. That is the final answer. Hope it helps!
During one month, a rental agency rented a total of 155 cars, trucks, and vans. Nine times as many cars were rented as vans, and three times as many vans were rented as trucks. Let x represent cars, let y represent vans and let z represent trucks. Write a system of three equations that represent the number of each vehicle rented
The system of equations that represents the number of each vehicle rented is: x + y + z = 155, x = 9y, and y = 3z.
Explanation:The question represents a system of linear equations. With the agreed notations: Let x represent cars, let y represent vans and let z represent trucks. We are given that:
The total number of all vehicles rented was 155. Therefore, the first equation is: x + y + z = 155. It was also given that nine times as many cars were rented as vans. Thus, the second equation is: x = 9y. Finally, three times as many vans were rented as trucks, giving us the third equation: y = 3z.Learn more about Linear Equations here:
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What is the height of the pyramid?
Answer:
Step-by-step explanation:
280 cubit
Answer:
Step-by-step explanation:
The formula for the volume,V of the square base pyramid is
V = 1/3(lwh)
Where
l = length of one side of the base of the pyramid.
w = length of the other side of the base of the pyramid.
h = the perpendicular height of the pyramid. Since the base of the pyramid is a square, l = w
The Volume, V is given as 18069333.3333 cubits^3
l = 440 cubits
w = 440 cubits
18069333.3333 = 1/3 × 440 × 440 × h
18069333.3333 = 64533.33h
h = 18069333.3333/64533.33
h = 280 cubits
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Which of the following are the vertices of the image of the figure below under the translation (x , y) ---> (x + 4, y - 2)? SELECT ALL THAT APPLY
A. (- 1, 2)
B. (- 9, 6)
C. (- 2, 7)
D. (6, 3)
E. (1, - 4)
F. (- 7, 0)
Answer:
d b f
Step-by-step explanation:
Mae king earns a weekly salary of $305 plus a 7.5% commission on sales at a gift shop.How much would she make in a work week if she sold 4,300 worth of merchandise
Answer:
Step-by-step explanation:
Mae king earns a weekly salary of $305 plus a 7.5% commission on sales at a gift shop. This means that the total amount that she can earn in a week is not fixed. If in a week, she sold 4,300 worth of merchandise, her commission on this amount of sales will be 7.5 % of 4,300
Commission on sales = 7.5/100× 4300 = 0.075×4300= $332.25
Amount of money made for the week will be the sum of her weekly salary and the commission earned on sales. It becomes
305 + 332.25 = $627.5
Please help me with this!!!
Answer:
{A, H, M, O, P, R, S, T} = {1, 7, 5, 0, 8, 6, 4, 9}
or
{O, A, S, M, R, H, P, T} = {0, 1, 4, 5, 6, 7, 8, 9}
Step-by-step explanation:
Starting in the thousands column, we see the sum P+M+A mod 10 = M, so P + A = 10 or 11. That is, there is a carry to the next column of 1, meaning T + 1 = O, and that sum must also create a carry of 1, so S + 1 = M.
In order for T + 1 to generate a carry, we must have T = 9 and O = 0.
Now, consider the 10s column. This has 36 +A +(carry in) mod 10 = 9. So, A+(carry in) = 3.
Considering the 1s column, we have 9+0+2H+S = H+10 or H+20. We know H+S+9 cannot be 10, so it must be 20. That means H+S = 11, and (carry in) to the 10s column must be 2. Since A = 3 - (carry in), we must have A=1.
At this point, we have ... A=1, T=9, O=0, S+H=11, S+1=M.
Now, consider the 100s column. We know the carry in from the 10s column is 3, so we have 3+2A+R=A+10. Since we know A=1, this means 5+R=11, or R=6.
The carry in to the 1000s column is 1, so we have P+A+1 = 10, or P=8.
__
Our assignments so far are ...
0 = O, 1 = A, 6 = R, 8 = P, 9 = T.
and we need to find S, M, and H such that M=S+1 and S+H=11. We know S and H cannot be 2, 3, or 5, because the 11's complement of those digits is already assigned. That leaves 4 and 7 for S and H, but we also need an unassigned value that is 1 more than S. These considerations make it necessary that S=4, M=5, H=7.
Then the addition problem is ...
8197 + 90 + 5197 +491694 +19 = 505197
_____
Final assignments are ...
O = 0, A = 1, S = 4, M = 5, R = 6, H = 7, P = 8, T = 9
A car repair shop offers its customers free coffee while they wait. By the end of each day, the coffee urn, which had started out with 7 1/4 gallons of coffee, was left with 2 1/12 gallons. How many gallons of coffee had been dispensed?
Answer: 5.1667 gallons of coffee had been dispensed
Step-by-step explanation:
Let x represent the number of gallons of coffee that had been dispensed.
By the end of each day, the coffee urn, which had started out with 7 1/4 gallons of coffee, was left with 2 1/12 gallons. This means that the initial number of gallons of coffee was 7 1/4 = 7.25 gallons
The amount left after dispense x gallons is 2 1/12 = 2.0833 gallons.
Therefore,
x +2.0833 = 7.25
x = 7.25 - 2.0833 = 5.1667 gallons
Use synthetic division to check that the number k is a lower bound for the real zeros of the function f.
k=-7; f(x)= 5x^3 - 8x^2 + x -5
Answer:
see below
Step-by-step explanation:
The signs on the bottom line alternate, so the value of k is, indeed, a lower bound.
_____
Comment on lower bound for this cubic
The signs of the coefficients alternate, so Descartes' rule of signs will tell you there are zero negative real roots. That is, 0 is a lower bound for real roots. No synthetic division is needed.
Elizabeth brought a box of donuts to share. There are two-dozen (24) donuts in the box, all identical in size, shape, and color. Three are jelly-filled, four are lemon-filled, and seventeen are custard-filled. You randomly select one donut, eat it, and select another donut.
Find the probability of selecting a lemon-filled donut followed by a custard-filled donut.
Answer:
P = 68/552 = 0.123 or 12.3%
Step-by-step explanation:
First, let's calculate the probability of getting a lemon donut. We have only 4 lemon donut among 24 donuts, so probability is:
P(A) = 4/24
Next, as we already ate the lemon donut, we only have 23 donuts now, and among these 23, 17 are custard filled, so probability of choosing one of those is:
P(B) = 17/23
But we want to know the probability that the custard filled donut is choosen after you eat the lemon one so:
P(B|A) = P(A) * P(B)
Replacing:
P(B|A) = 4/24 * 17/23
P(B|A) = 68/552 = 0.123 or 12.3%
Julie had three wheels from bikes and things that she stacked against the shed. Each wheel fitted so neatly together that Julie took this photo. The radius of the largest sheet is 16 cm and the radius of the middle-sized wheel is 9cm. What is the radius of Julie's smallest wheel?
Answer:
12.5 cm
Step-by-step explanation:
16 - 9 = 7
7 (1/2) = 3.5
9 + 3.5 = 12.5
15 - 3.5 = 12.5
Machines A and B always operate independently and at their respective constant rates. When working alone, Machine A can fill a production lot in 5 hours, and Machine B can fill the same lot in x hours. When the two machines operate simultaneously to fill the production lot, it takes them 2 hours to complete the job. What is the value of x ?
Answer:
The value of x is [tex]\frac{10}{3}[/tex] hours.
Step-by-step explanation:
Machine A = 5 hours
Machine B = x hours
Machine A and B = 2 hours
Using the formula: [tex]\frac{T}{A} + \frac{T}{B} = 1[/tex]
where:
T is the time spend by both machine
A is the time spend by machine A
B is the time spend by machine B
[tex]\frac{2}{5} + \frac{2}{x} = 1[/tex]
Let multiply the entire problem by the common denominator (5B)
[tex]5x(\frac{2}{5} + \frac{2}{x} = 1)[/tex]
2x + 10 = 5x
Collect the like terms
10 = 5x - 2x
10 = 3x
3x = 10
Divide both side by the coefficient of x (3)
[tex]\frac{3x}{3} = \frac{10}{3}[/tex]
[tex]x = \frac{10}{3}[/tex] hours.
Therefore, Machine B will fill the same lot in [tex]\frac{10}{3}[/tex] hours.
What is the solution of the linear-quadratic system of equations?
y=x^2+5 −3
y − x = 2
The answer is (1,-5).
The solution to the linear-quadratic system of equations is found by substituting y from the linear equation into the quadratic equation and solving for x, then back-solving for y. The system has two solutions: (-5, -3) and (1, 3).
Explanation:To solve the linear-quadratic system of equations:
First, let's use substitution. The second equation can be rearranged to y = x + 2. Substituting this into the first equation gives us:
x + 2 = x2 + 5x - 3
Let's move all terms to one side to make it a quadratic equation:
x2 + 5x - x - 3 - 2 = 0
x2 + 4x - 5 = 0
This is a quadratic equation that can be factored into:
(x + 5)(x - 1) = 0
Setting each factor equal to zero gives us the solutions for x:
Now we'll substitute these x-values back into y = x + 2 to find the corresponding y-values:
Therefore, the system has two solutions: (-5, -3) and (1, 3).
If we needed to use the quadratic formula, it would be in the context of an equation that is not easily factorable.
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Jill planted two flowers in her garden. The first flower is 2 inches tall, and it is growing 2.25 inches each week. The second plant is 5.75 inches tall, and it is growing 1.5 inches each week. How many weeks will it be until the two plants are the same height?
Answer:it will take the two plants 6 weeks before the heights are the same
Step-by-step explanation:
Jill planted two flowers in her garden.
The first flower is 2 inches tall, and it is growing 2.25 inches each week. Since the growth rate is in an arithmetic progression, we will apply the formula for finding the nth term of the series
Tn = a + (n - 1)d
Tn = the nth height of the first flower
a = the initial height of the first flower
d = the common difference in height of the first flower weekly
n = number of weeks
From the information given,
For the first flower,
a = 2
d = 2.25
Tn ?
n ?
Tn = 2 + (n - 1)2.25
For the second flower,
a = 5.75
d = 1.5
Tn ?
n ?
Tn = 5.75 + (n - 1)1.5
To determine the number of weeks that it will take until the two plants are the same height, we would equate Tn for both flowers. It becomes
2 + (n - 1)2.25 = 5.75 + (n - 1)1.5
2 + 2.25n - 2.25 = 5.75 + 1.5n - 1.5
Collecting like terms
2.25n - 1.5n = 5.75 - 1.5 - 2 + 2.25
0.75n = 4.5
n = 4.5/0.75
n = 6 weeks
To find out how many weeks it will be until the two plants are the same height, set up an equation and solve for x. The plants will be the same height after 5 weeks.
Explanation:To find out how many weeks it will be until the two plants are the same height, we need to set up an equation. Let the number of weeks be represented by x. The height of the first plant can be represented as 2 + 2.25x, and the height of the second plant can be represented as 5.75 + 1.5x. Set these two expressions equal to each other: 2 + 2.25x = 5.75 + 1.5x.
To solve for x, subtract 1.5x from both sides: 2 + 0.75x = 5.75.
Then, subtract 2 from both sides: 0.75x = 3.75.
Finally, divide both sides by 0.75 to solve for x: x = 5.
Solve the inequality and graph the solution
|2x - 7| > 1
Answer:
Open circle to the right of 4
x > 4
Step-by-step explanation:
Add 7 to both sides
2x > 8
x > 4
Open circle to the right of 4
Suppose that the functions r and a are defined for all real numbers x as follows. r(x)=2x-1 S(x)=5x write the expressions for (r-s)(x)and(r•s)(x)and evaluate(r+s)(-2).
[tex]\boxed{(r-s)(x)=-3x-1} \\ \\ \boxed{(r\cdot s)(x)=10x^2-5x} \\ \\ \boxed{(r+s)(-2)=-15}[/tex]
Explanation:In this exercise, we have the following functions:
[tex]r(x)=2x-1 \\ \\ s(x)=5x[/tex]
And they are defined for all real numbers x. So we have to write the following expressions:
First expression:
[tex](r-s)(x)[/tex]
That is, we subtract s(x) from r(x):
[tex](r-s)(x)=2x-1-5x \\ \\ Combine \ like \ terms: \\ \\ (r-s)(x)=(2x-5x)-1 \\ \\ \boxed{(r-s)(x)=-3x-1}[/tex]
Second expression:
[tex](r\cdot s)(x)[/tex]
That is, we get the product of s(x) and r(x):
[tex](r\cdot s)(x)=(2x-1)(5x) \\ \\ By \ distributive \ property: \\ \\ (r\cdot s)(x)=(2x)(5x)-(1)(5x) \\ \\ \boxed{(r\cdot s)(x)=10x^2-5x}[/tex]
Third expression:
Here we need to evaluate:
[tex](r+s)(-2)[/tex]
First of all, we find the sum of functions r(x) and s(x):
[tex](r+s)(x)=2x-1+5x \\ \\ Combine \ like \ terms: \\ \\ (r+s)(x)=(2x+5x)-1 \\ \\ (r+s)(x)=7x-1[/tex]
Finally, substituting x = -2:
[tex](r+s)(-2)=7(-2)-1 \\ \\ (r+s)(-2)=-14-1 \\ \\ \boxed{(r+s)(-2)=-15}[/tex]
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Robert's father is 4 times as old as robert. After 5 years, father will be three times as old as robert.What is their present ages of robert and his father respectively
Answer: Robert's present age is 10 years
Robert father's present age is 40 years
Step-by-step explanation:
Let r = Robert's current age
Let y = Robert father's current age
Robert's father is 4 times as old as robert. This means that
y = 4x
After 5 years, Robert's father will be three times as old as Robert. This means that
y + 5 = 3(x+5)
y + 5 = 3x + 15 - - - - - - - ;1
We will substitute y = 4x into equation 1. It becomes
4x + 5 = 3x + 15
Collecting like terms,
4x - 3x = 15 - 5
x = 10
y = 4x
Substituting x = 10,
y = 4× 10 = 40 years