Answer:
x/4 + 3/28 = y
Step-by-step explanation:
To find the inverse, switch the x's and y's (note that f(x) is y) and solve for y:
x = 4y - 3/7
x + 3/7 = 4y
x/4 + 3/28 = y
Answer:
−1(x) = 3√2(x+7) 2 f - 1 (x) = 2 (x + 7) 3 2 is the inverse of f (x) = 4x3 − 7 f (x) = 4 x 3 - 7.
During the period 2000-2016, the average costs B (in dollars) for a new boat and the average costs H (in dollars) for a hovercraft can be modeled
by the following functions:
B = 30 +140 and H=-1012 +350 + 400
where t is the number of years since 2000
(a) How can the difference of the costs of a hovercraft and the costs of a boat be expressed? (4 points)
(b) About how much was the difference in the year 2010? (2 points)
The difference between the costs of a hovercraft and a boat in the year 2010 was approximately $1,558.
(a) The difference in costs between a hovercraft and a boat can be expressed as follows:
Cost difference = Hovercraft cost - Boat cost
Cost difference = (-1012 + 350t + 400) - (30 + 140t)
Cost difference = -1012 + 350t + 400 - 30 - 140t
Cost difference = 220t - 642
Therefore, the difference in costs between a hovercraft and a boat can be expressed as 220t - 642.
(b) To find the difference in the year 2010, we need to substitute t = 10 in the expression we derived in part (a):
Cost difference = 220t - 642
Cost difference = 220(10) - 642
Cost difference = 2200 - 642
Cost difference = 1558
Therefore, the difference between the costs of a hovercraft and a boat in the year 2010 was approximately $1,558.
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Find the length of PQ.
Assume that lines which appear tangent are tangent.
Applying the secant-tangent theorem, the length of PQ is calculated as: 6 units.
How to Find the Length Using the Secant-Tangent Theorem?In the image given, line segment QS is a secant while QP is a tangent. This, according to the secant-tangent theorem, we have:
(x - 3)(x - 3 + 5) = (x - 1)²
(x - 3)(x + 2) = (x - 1)(x - 1)
Expand:
x² - x - 6 = x² - 2x + 1
Combine like terms:
x² - x² - x + 2x = 6 + 1
x = 7
length of PQ = x - 1
Plug in the value of x:
length of PQ = 7 - 1 = 6 units.
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Qn in attachment. ..
Answer:
pls mrk me brainliest (´(ｪ)｀）
Tim can paint a room for 6 hours. Bella can paint the same room for 4 hour. How many hours would it take Tim and Bella to paint the room while working together?
Tim and Bella working together can paint the room in 2.4 hours.
We can use the formula:
1/Total Work = 1/Time1 + 1/Time2
where Total Work is the work to be done (in this case, painting the room), Time1 is the time taken by Tim to complete the work, and Time2 is the time taken by Bella to complete the work.
Let x be the time taken by Tim and Bella working together to complete the work. Then we can write:
1/1x = 1/6 + 1/4
Simplifying this equation, we get:
1/1x = 5/12
Multiplying both sides by x, we get:
x = 12/5 = 2.4 hours.
Therefore, Tim and Bella working together can paint the room in 2.4 hours.
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salvador recorded in this list the heights in millimeters of each of his bean plants.
52, 46, 51, 32,50
which 2 inequalities best describe, h, the plant heights in millimeters?
h < 32, h > 52
h> 32, h < 52
h < 46, h > 52
h < 46, h > 52
The two inequalities that best describe the plant heights in millimeters are: h > 32 and h < 52. This is because all the recorded heights fall within this range. The other options do not include all the recorded heights or include heights that are not recorded.
To find the best inequalities that describe the plant heights (h) in millimeters, we need to determine the minimum and maximum heights from the given list.
List of plant heights: 52, 46, 51, 32, 50
Minimum height: 32 mm
Maximum height: 52 mm
Now we can write the inequalities that best describe the plant heights:
h > 32 (heights are greater than 32 mm)
h < 52 (heights are less than 52 mm)
Your answer: h > 32, h < 52
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moving company is using a large shipping container to pack up several smaller boxes. The smaller boxes are cubes with each side length measuring
The larger container will hold 20 smaller boxes along its width, 40 boxes along its length, and 20 boxes along its height.
e this information to fill in the blanks below.
please look at the photo:(
The volume of one smaller box is 1 cubic foot.
Solution to Shipping Container ProblemThe volume of one smaller box is calculated by raising the length of its sides to the power of 3, so the volume of one smaller box is:
Volume (smaller box) = (1 foot)³ = 1 cubic foot
The number of smaller boxes that fit into a shipping container is obtained by multiplying the number of boxes that fit along each dimension, so the number of smaller boxes that fit into a shipping container is:
Number of smaller boxes = 20 boxes (width) × 40 boxes (length) × 20 boxes (height) = 16,000 boxes
To calculate the volume of the large shipping container, we need to multiply the length, width, and height of the container. Since each dimension is given in terms of the number of boxes it can hold, we need to multiply each dimension by the length of one smaller box (1 foot) to obtain the dimensions in feet. Therefore, the volume of the large shipping container is:
Volume of the large shipping container = 20 boxes (width) × 1 foot (length of one smaller box) × 40 boxes (length) × 1 foot (length of one smaller box) × 20 boxes (height) × 1 foot (length of one smaller box) = 16,000 cubic feet
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Pls answer this, 5 points and brainliest for the one who answers first.
Answer: A
Step-by-step explanation:
It's A because our function of f is multiplied by 3.
Since our y intercept is 1, and we are multiplying the function f by 3,
our new y intercept is 3, meaning it is A.
Another way to check this is by using the other two points on your graph.
Please give brainliest + have a good afternoon.
Answer:
Step-by-step explanation:
Your original function has points at
0, 1
1, 2
2,4
if you stretched it by 3, multiply your y by 3
new function:
0, 3
1, 6
2, 12
Jenny and Kera are playing a game. Jenny has -10 points and loses 5 more points. How many points does Jenny have now? Kera has 22 points and loses 14 points. How many points does Kera have now? Make sure that you type each name with her current score
The current scores of both Jenna and Kera in the game they were playing are
Jenny's current score is -15.
Kera's current score is 8.
How many points does Jenny have now?In the given problem, we are given two players, Jenny and Kera, playing the game.
Jenny has -10 points, which means she already has negative points. He has since lost 5 more points. So his current score would be:
-10 - 5 = -15
So now Jenny has a score of -15.
Kera, meanwhile, has 22 points and 14 points to lose. So his current score would be:
22 - 14 = 8
So now Kera has 8 points.
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Ginny's family traveled
3
10
of the distance to her aunt’s house on Saturday. They traveled
3
7
of the remaining distance on Sunday. What fraction of the total distance to her aunt’s house was traveled on Sunday?
Ginny's family traveled 3/10 of the total distance to her aunt's house on Sunday.
What fraction of the total distance was traveled on Sunday?Let's say the total distance to Ginny's aunt's house is represented by the fraction 1.
On Saturday, Ginny's family traveled 3/10 of the distance, leaving 7/10 of the distance remaining.
Then, on Sunday, they traveled 3/7 of the remaining distance.
To find out what fraction of the total distance was traveled on Sunday, we need to multiply the distance traveled on Saturday and Sunday and divide by the total distance:
So, we have
Fraction = 3/7 * 7/10
Evaluate
Fraction = 3/10
Therefore, Ginny's family traveled 3/10 of the total distance to her aunt's house on Sunday.
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Last year at a certain high school, there were 124 boys on the honor roll and 125 girls on the honor roll. This year, the number of boys on the honor roll decreased by 25% and the number of girls on the honor roll decreased by 20%. By what percentage did the total number of students on the honor roll decrease? Round your answer to the nearest tenth (if necessary).
Answer:
22.5%
Step-by-step explanation:
Last year, there were 124 boys and 125 girls, meaning 249 total.
This year, if boys decreased by 25%:
[tex]0.25 * 124 = 31.[/tex]
A decrease of 31 boys.
If girls decreased by 20%:
[tex]0.2 * 125 = 25.[/tex]
A decrease of 25 girls.
If there was a total decrease of 56 students from last year to this year, the total decrease is:
[tex]56/249*100=22.5.[/tex]
A decrease of 22.5%.
Find the limit of (7x3)/(4x2-2x+10) as x approaches infinity."
To find the limit of (7x3)/(4x2-2x+10) as x approaches infinity, we need to divide the highest power of x in the numerator and denominator, which is x3, by the highest power of x in the denominator, which is x2. This gives us: (7x3)/(4x2-2x+10) = (7/4)x
As x approaches infinity, the value of (7/4)x also approaches infinity. Therefore, the limit of (7x3)/(4x2-2x+10) as x approaches infinity is infinity.
To find the limit of (7x^3)/(4x^2-2x+10) as x approaches infinity, we'll first look at the highest powers of x in the numerator and denominator.
In this case, the highest power of x in the numerator is x^3, and in the denominator, it's x^2. Since the highest power of x in the numerator is greater than that in the denominator, the limit will go to infinity (or -infinity) depending on the coefficients of the highest powers.
For this function, the coefficients are positive (7 for x^3 and 4 for x^2), so the limit as x approaches infinity will be positive infinity.
Your answer: The limit of (7x^3)/(4x^2-2x+10) as x approaches infinity is positive infinity.
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A math teacher gave her class two tests. The results were 42 percent of the class passed the first test and 50 percent of the class passed thesecond test. Also, 60 percent of the students in the class who passed the first test passed the second test. What percentage of the class passed both tests? Are the events the class passed the first test and the class passed the second test independent? (1 point)
A) A total of 30 percent passed both tests. The events are independent.
B) A total of 25. 2 percent passed both tests. The events are independent.
C) A total of 25. 2 percent passed both tests. The events are not independent.
D) A total of 30 percent passed both tests. The events are not independent.
The answer is option C) A total of 25.2 percent passed both tests. The events are not independent.
To solve this problem, we can use a Venn diagram. Let P(A) be the probability of passing the first test, P(B) be the probability of passing the second test, and P(A and B) be the probability of passing both tests.
From the problem, we know:
P(A) = 0.42
P(B) = 0.50
P(B | A) = 0.60 (the probability of passing the second test given that the student passed the first test)
We can use the formula P(B | A) = P(A and B) / P(A) to find P(A and B):
0.60 = P(A and B) / 0.42
P(A and B) = 0.60 x 0.42
P(A and B) = 0.252
Therefore, 25.2% of the class passed both tests.
To determine if the events are independent, we can compare P(B) to P(B | A). If they are equal, the events are independent. If they are not equal, the events are dependent.
P(B) = 0.50
P(B | A) = 0.60
Since P(B) is not equal to P(B | A), the events are dependent.
Therefore, the answer is option C) A total of 25.2 percent passed both tests. The events are not independent.
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WORK OUT THE SIZE OF AN EXTERIOR ANGLE OF A REGULAR HEXAGON
Answer:
60°
Step-by-step explanation:
A hexagon has 6 angles
The sum of the measures of the exterior angles of a hexagon is equal to 360°
So, measure of each exterior angle = 360∘ / 6 = 60∘
Two random samples 4. 49, 7. 68, 5. 97, 0. 97, 6. 88, 6. 07, 03. 08, 04. 02, 03. 83, 6. 35, and 4. 59, 3. 39, 3. 79, 6. 89, 5. 07, 07. 41, 0. 44, 2. 47, 4. 80, 7. 23 were obtained independently from distributions with the same mean. Perform a permutation test to test the hypothesis that the variability in both populations is the same against the alternative that it is larger in the second population. As a test statistic use: (i) The difference of sample ranges. (ii) The ratio of sample variances. (iii) Compare both results
A. Main Answer:
The hypothesis that the variability in both populations is the same against the alternative that it is larger in the second population, we can perform a permutation test using two different test statistics:
(i) the difference of sample ranges and
(ii) the ratio of sample variances. By comparing the results of both test statistics, we can draw conclusions about the variability in the populations.
(i) Difference of Sample Ranges:
1. Calculate the sample range for each sample. The sample range is the difference between the maximum and minimum values in the sample.
Sample 1 Range = Maximum value - Minimum value for Sample 1
Sample 2 Range = Maximum value - Minimum value for Sample 2
2. Calculate the observed difference of sample ranges, which is the difference between the sample range of Sample 2 and Sample 1.
3. Pool the data from both samples and shuffle them randomly, keeping the same sample sizes.
4. Calculate the difference of sample ranges for the shuffled data.
5. Repeat steps 3 and 4 many times (e.g., 1000 permutations) to obtain a distribution of the difference of sample ranges under the null hypothesis (where variability is the same in both populations).
6. Compare the observed difference of sample ranges from step 2 with the distribution obtained from the permutations.
(ii) Ratio of Sample Variances:
1. Calculate the sample variance for each sample.
2. Calculate the observed ratio of sample variances, which is the ratio of the sample variance of Sample 2 to the sample variance of Sample 1.
3. Pool the data from both samples and shuffle them randomly, keeping the same sample sizes.
4. Calculate the ratio of sample variances for the shuffled data.
5. Repeat steps 3 and 4 many times (e.g., 1000 permutations) to obtain a distribution of the ratio of sample variances under the null hypothesis.
6. Compare the observed ratio of sample variances from step 2 with the distribution obtained from the permutations.
By comparing the results of both test statistics, we can assess whether the variability in the second population is significantly larger than the first population. If both test statistics consistently indicate larger variability in the second population, it provides evidence against the null hypothesis and suggests that the variability in the second population is indeed larger.
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Mark invests $5,000 and earns $375 in simple interest over a 3 year period. What was the interest rate on the investment? answer is 2.5%
Three questions in this section. Answer all questions in this section on OnQ. You do not need to submit solutions to questions in this section. Question 1 (1 point) Suppose the point (0, -1) is a critical point of the function f(x,y) = x3 + y3 – 6x2 – 3y – 5. - Which one of the following statements is true? The point (0, -1) is a global maximum of f(x, y) The point (0, -1) is a local minimum of f(x, y) The point (0, -1) is a local maximum of f(x, y) The point (0, -1)is a saddle point of f(x, y)
The point (0, -1) is a saddle point of f(x, y).
To determine the nature of the critical point (0, -1) for the function f(x, y) = x^3 + y^3 - 6x^2 - 3y - 5, we need to use the second partial derivative test. First, we compute the second partial derivatives:
f_xx = 6x - 12
f_yy = 6y
f_xy = f_yx = 0
Now, evaluate these at the critical point (0, -1):
f_xx(0, -1) = -12
f_yy(0, -1) = -6
f_xy(0, -1) = 0
Calculate the determinant D = f_xx * f_yy - f_xy^2:
D = (-12) * (-6) - 0^2 = 72
Since D > 0 and f_xx < 0 at the critical point, we can conclude that (0, -1) is a local maximum of f(x, y).
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Gross Monthly Income: Jackson works for a pipe line company and is paid $18. 50 per hour. Although he will have overtime, it is not guaranteed when or where, so Jackson will only build a budget on 40 hours per week. What is Jackson’s gross monthly income for 40 hours per week? Type in the correct dollar amount to the nearest cent. Do not include the dollar sign or letters.
A. Gross Annual Income: $
B. Gross Monthly Income: $
To find Jackson's gross monthly income for 40 hours per week, follow these steps:
1. Calculate his weekly income: Multiply his hourly wage ($18.50) by the number of hours he works each week (40 hours).
2. Calculate his monthly income: Multiply his weekly income by the number of weeks in a month (4 weeks).
A. Gross Annual Income: To find this, multiply his monthly income by 12 (the number of months in a year).
B. Gross Monthly Income: This is the answer we need to find.
Step-by-step calculations:
1. Weekly income = $18.50 * 40 hours = $740
2. Monthly income = $740 * 4 weeks = $2,960
A. Gross Annual Income: $2,960 * 12 = $35,520
B. Gross Monthly Income: $2,960
The answer:
A. Gross Annual Income: $35,520
B. Gross Monthly Income: $2,960
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Verify that the sample standard deviations use of ANO\A allow the the means to compare the population means. What do the suggest about the effect of the subject’s gender and attractiveness of the confederate on the evaluation of the product?
On performing an ANOVA test the p-value obtained is less than the chosen significance level, hence it is verified that the sample standard deviations use of ANOVA allow the the means to compare the population means. Since ANOVA test shows significant difference therefore, it suggests that these factors play a role in influencing the evaluation of the product.
To verify that the sample standard deviations use of ANOVA allows means to compare the population means, discuss the terms ANOVA, sample standard deviation, population means.
1. ANOVA (Analysis of Variance): ANOVA is a statistical method used to compare the means of multiple groups to determine if there's a significant difference between them.
2. Sample Standard Deviation: Sample standard deviation is a measure of how spread out the values in a sample are. It helps estimate the population standard deviation, which is necessary for calculating the F statistic in ANOVA.
a. Calculate the sample means and standard deviations for each group.
b. Perform an ANOVA test using calculated means and standard deviations.
c. Interpret results: If p-value obtained from the ANOVA test is less than the chosen significance level (e.g., 0.05), it means there is a significant difference between population means.
Regarding the effect of the subject's gender and attractiveness of the confederate on the evaluation of the product, if the ANOVA test shows a significant difference, it suggests that these factors play a role in influencing the evaluation of product. You can further analyze the data by performing post-hoc tests to identify which specific groups differ significantly.
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Of the last 12 cakes sold at graces cakes,6 were carrot cakes. Find the experimental probability the next cake sold will be a carrot cake.in percentages.
Wes has 20 feet of garden fencing. If he
wants the smallest side of his garden
to be 3 feet or longer, what possible
rectangles can he make?
The possible rectangles that Wes can make with his 20 feet of fencing are:
L = 3, W = 7
L = 4, W = 6
L = 5, W = 5
L = 6, W = 4
L = 7, W = 3
How to find the possible rectangles?Let L be the length of the rectangular garden and W be the width of the garden. Since the garden is enclosed by four sides, Wes will need 2L+2W feet of fencing to enclose it. We know that he has 20 feet of fencing, so we have the equation:
2L + 2W = 20
We also know that the smallest side of the garden should be 3 feet or longer, so:
L >= 3
W >= 3
To find the possible rectangles Wes can make, we can solve the equation for one variable in terms of the other:
2L + 2W = 20
2L = 20 - 2W
L = 10 - W
Now we can substitute this expression for L into the inequality L >= 3 to get:
10 - W >= 3
W <= 7
Similarly, we can substitute L = 10 - W into the inequality W >= 3 to get:
10 - L >= 3
L <= 7
Therefore, the possible values for L and W are:
3 <= L <= 7
3 <= W <= 7
We can also use the equation 2L + 2W = 20 to find the combinations of L and W that add up to 10, since the total length of fencing is 20 feet:
L = 3, W = 7
L = 4, W = 6
L = 5, W = 5
L = 6, W = 4
L = 7, W = 3
These are the possible rectangles that Wes can make with his 20 feet of fencing, where the smallest side is 3 feet or longer.
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You're at a clothing store that dyes your clothes while you wait. The store offers
4 different articles of clothing and
3 colors.
If you randomly choose the article of clothing and the color, which of these diagrams can be used to find all of the possible outcomes?
The probability of randomly selecting an orange hat is 1/12 or 0.0833.
What is the probability of randomly selecting an orange hat?In the sample, there are total of 4 pieces of clothing and 3 colors.
The possible outcomes is:
= 4 x 3
= 12
So, the outcomes when randomly selecting a piece of clothing and a color is 12.
From 12 possible outcomes, there is only one outcome where you end up with an orange hat.
So, the probability of randomly selecting an orange hat is:
= 1/12
= 0.0833.
Correct question "You're at a clothing store that dyes your clothes while you wait. You get to pick from 4 pieces of clothing (shirt, pants, socks, or hat) and 3 colors (purple, blue, or orange). If you randomly pick the piece of clothing and the color, what is the probability that you'll end up with an orange hat?"
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2. 7.G.1.2 Can a quadrilateral be drawn that meets the conditions described below? Select Yes or No by placing a check or X in the appropriate box. Conditions Two pairs of parallel sides and at least two right angles One pair of parallel sides and no right angles One pair of parallel sides and three right angles No parallel sides and four right angles Yes No
The complete conditions are
Two pairs of parallel sides and at least two right angles : YesOne pair of parallel sides and no right angles :YesOthers are NoChecking if a quadrilateral can be drawn from the conditionsBy definition a quadrilateral is a shape that has four sides and four angles
Next, we test the conditions
Two pairs of parallel sides and at least two right angles
This is true because quadrilaterals like rectangles and squares have two pair of parallel sides and right angles
One pair of parallel sides and no right angles
This is also true because quadrilaterals like trapezoid have one pair of parallel sides and may or may not have right angle
One pair of parallel sides and three right angles
This is false because a quadrilateral cannot be drawn with this condition
No parallel sides and four right angles
This is false because a quadrilateral cannot be drawn with this condition
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Find the first four nonzero terms of the Taylor series for the function cos (20²) about 0. f(0) = NOTE: Enter only the first four non-zero terms of the Taylor series in the answer field. Coefficients must be exact. f(0) = Find the first four nonzero terms of the Taylor series for the function f(y) = ln(1 - 4y¹) about 0. f(y) NOTE: Enter only the first four non-zero terms of the Taylor series in the answer field. Coefficients must be exact. = +.. +...
The first four nonzero terms for cos(20x²) are:
1
The first four nonzero terms for ln(1 - 4y) are:
-4y + 8y² - 32y³ +...
Taylor series:
To find the first four nonzero terms of the Taylor series for the function cos(20x²) about 0,
we need to find the first few derivatives of the function, and evaluate them at x = 0.
f(x) = cos(20x²)
f'(x) = -40x * sin(20x²)
f''(x) = -40(40x² * cos(20x²) + 20sin(20x²))
f'''(x) = 40(1600x³ * sin(20x²) + 120x * cos(20x²))
Now, evaluate these at x = 0:
f(0) = cos(0) = 1
f'(0) = 0 (since sin(0) = 0
f''(0) = -40(0) = 0
f'''(0) = 0 (since cos(0) = 1
The first four nonzero terms for cos(20x²) are:
1
Now, let's find the first four nonzero terms of the Taylor series for the function f(y) = ln(1 - 4y) about 0.
f(y) = ln(1 - 4y)
f'(y) = -4 / (1 - 4y)
f''(y) = 16 / (1 - 4y)²
f'''(y) = -96 / (1 - 4y)³
Evaluate these at y = 0:
f(0) = ln(1) = 0
f'(0) = -4 / (1) = -4
f''(0) = 16 / (1)² = 16
f'''(0) = -96 / (1)³ = -96
The first four nonzero terms for ln(1 - 4y) are:
-4y + 8y² - 32y³ +...
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Please help! This is part of my grade, please make sure to read the question before answering because I need this to be correct (35 points)
Answer:
u = -2.34 or u = 18.34
Step-by-step explanation:
You want to solve u² -16u = 43 by completing the square.
Completing the squareTo complete the square, add the square of half the coefficient of the linear term to both sides.
u² -16u +(-16/2)² = 43 +(-16/2)²
u² -16u +64 = 107 . . . . . . . simplify
(u -8)² = 107 . . . . . . . . . write as a square
u -8 = ±√107 . . . . . . square root
u = 8 ± √107 . . . add 8
u = -2.34 or u = 18.34 . . . . . find the decimal values
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One salt solution is 20% salt and another is 60% salt. How many cubic centimeters of each solution must be mixed to obtain 100 cubic centimeters of a 30% salt solution?
Answer: Let's denote the number of cubic centimeters of the 20% salt solution as x and the number of cubic centimeters of the 60% salt solution as y.
We know that the total volume of the mixture is 100 cubic centimeters, so we have:
x + y = 100
We also know that the final solution should be a 30% salt solution. This means that the amount of salt in the final solution should be 0.3 times the total volume of the solution:
0.3(100) = 0.20x + 0.60y
where 0.20x represents the amount of salt in the 20% salt solution and 0.60y represents the amount of salt in the 60% salt solution.
We now have two equations with two unknowns:
x + y = 100
0.20x + 0.60y = 30
We can solve for x and y by using any method of linear equations, such as substitution or elimination.
Here, we will use substitution. Solving the first equation for x, we get:
x = 100 - y
Substituting this expression for x in the second equation, we get:
0.20(100 - y) + 0.60y = 30
Simplifying and solving for y, we get:
20 - 0.20y + 0.60y = 30
0.40y = 10
y = 25
So, we need 25 cubic centimeters of the 60% salt solution.
To find the amount of the 20% salt solution, we can substitute this value of y back into either equation:
x + y = 100
x + 25 = 100
x = 75
So, we need 75 cubic centimeters of the 20% salt solution.
Therefore, we need to mix 75 cubic centimeters of the 20% salt solution and 25 cubic centimeters of the 60% salt solution to obtain 100 cubic centimeters of a 30% salt solution.
Omar cuts a piece of wrapping paper with the shape and dimensions as shown.Find The Area Of The Wrapping Paper.Round Your Answer To The Nearest Tenth If Needed
The total area of the wrapping paper is 72.5 in².
In the given figure (attached below), we have two shapes one is a triangle and the other one is a rectangle. To find the total area of the wrapping paper we have to add the area of the rectangle part and the area of the trianglular part.
Total area = Area of the rectangular part + area of the triangular part.
Area of the rectangular part = length x breadth
from the below figure, length = 15 in
breadth = 4 in
So, area of the rectangular part = 15 in x 4 in = 60 in²
Similarly, area of the triangular part = 1/2 x base x height
from the below figure, base of the triangle = 15 in -10 in = 5 in
height of the triangle = 9 in - 4 in = 5 in
So, area of the triangular part = 1/2 x 5 in x 5in = 12.5 in²
Now, the total area of the wrapping paper = area of the rectangular part + area of the triangular part = 60 in² + 12.5 in² = 72.5 in².
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Create a bucket by rotating around the y axis the curve y = 4 ln(x - 4) from y = 0 to y = 3. If this bucket contains a liquid with density 860 kg/m filled to a height of 2 meters, find the work required to pump the liquid out of this bucket (over the top edge). Use 9.8 m/s2 for gravity. Work = Preview Joules License Points possible: 1 This is attempt 1 of 3.
The work required to pump the liquid out of the bucket is approximately 2.482 x 10⁷ Joules.
How to find the work requiredTo create a bucket by rotating around the y-axis, we will use the formula for volume of revolution:
V = π ∫[a,b] (f(y))² dy where f(y) is the function being rotated, and a and b are the limits of integration.
In this case, the limits of integration are y = 0 and y = 3, and the function being rotated is y = 4 ln(x - 4), or x = e⁽y/⁴⁾ + 4.
So, we have:
V = π ∫[0,3] ((e⁽y/⁴⁾ + 4))² dy
V = π ∫[0,3] (e⁽y/²⁾ + 8e⁽y/⁴⁾+ 16) dy
V = π (2e⁽³/²⁾ + 32e⁽³/⁴⁾ + 48)
Now, to find the work required to pump the liquid out of the bucket, we need to use the formula:
W = ∫[h1,h2] ρgV(y) dy
where h1 is the height of the liquid (2 meters in this case), h2 is the height of the top edge of the bucket, ρ is the density of the liquid (860 kg/m^3), g is the acceleration due to gravity (9.8 m/s²), and V(y) is the volume of the liquid at height y.
To find V(y), we need to first find the radius of the bucket at height y.
The radius is given by: r(y) = e⁽y/⁴⁾+ 4
So, the volume of the liquid at height y is:
V(y) = π(r(y))² (h2 - y)
Plugging in the values, we have:
W = ∫[0,2] 860×9.8×π((e⁽y/⁴ + 4)²)×(2-y) dy
W = 2.482 x 10⁷J
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Please help!
For each problem approximate the area under the curve under the given interval using five trapezoids.
Answer:
area ≈ 9.219 square units
Step-by-step explanation:
You want the approximate area under the curve y = -1/2x² +x +5 on the interval [1.5, 4] using 5 trapezoids.
Trapezoid areaThe interval can be divided into 5 intervals of width ...
(4 -1.5)/5 = 2.5/5 = 0.5
The "bases" of each trapezoid will be the function values at the ends of the intervals, for example, at x=1.5 and x=2. The "height" of each trapezoid is the width of the sub-interval, 0.5.
The area formula for a trapezoid applies:
A = 1/2(b1 +b2)h
A = 1/2(f(x) +f(x +0.5))·0.5 . . . . . for x = 1.5, 2, 2.5, 3, 3.5
Approximate total areaThe sum of the areas is computed in the attachment as ...
area under the curve = 9.21875
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Additional comment
The value of the integral is 445/48 ≈ 9.2708333...
A rectangle with length n is inscribed in a circle of radius 9. Find an expression for the area of the rectangle in terms of n
The expression for the area of the rectangle inscribed in a circle of radius 9, in terms of n, is:
Area = n * √(81 - n^2)
To find an expression for the area of a rectangle inscribed in a circle of radius 9, we need to use the terms "rectangle" and "circle."
First, let's denote the length of the rectangle as "n" and the width as "w." Since the rectangle is inscribed in a circle with a radius of 9, its diagonal is equal to the diameter of the circle, which is 2 * 9 = 18.
Using the Pythagorean theorem for the right triangle formed by half of the diagonal, the length, and the width, we can write the equation:
(1/2 * 18)^2 = n^2 + w^2
81 = n^2 + w^2
Now, we need to express w in terms of n. To do this, we'll isolate w from the equation:
w^2 = 81 - n^2
w = √(81 - n^2)
The area of the rectangle can be calculated as the product of its length and width:
Area = n * w
Area = n * √(81 - n^2)
So, the expression for the area of the rectangle inscribed in a circle of radius 9, in terms of n, is:
Area = n * √(81 - n^2)
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If a triangle given by the matrix [235-102] is dialed by a scale factor of 2, what will will happened to the side lengths and angle measure of triangle?
If the triangle given by the matrix [235-102] is scaled by a factor of 2, the side lengths of the triangle will be doubled. The angle measures of the triangle will remain the same since scaling does not affect the angles.
When a triangle is scaled by a factor of 2, the side lengths will be doubled, but the angle measures will remain the same. Here's a step-by-step explanation:
1. The original matrix is [2 3 5; -1 0 2].
2. Apply the scale factor of 2 to each of the side lengths by multiplying the matrix by 2: [4 6 10; -2 0 4].
3. The side lengths of the triangle have been doubled, but the angle measures remain the same.
So, after scaling the triangle by a factor of 2, the side lengths will be doubled while the angle measures will stay the same.
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