a) A(5) = 10 * e^(-0.05*5)
b) 7.776
To solve the given problem, the initial amount of chlorine should be assumed as 'P0' and the exponential function should be written in the form of y = P_0 * e^(kt) where P_0 is the initial amount of chlorine and k is the rate of change. Therefore, for the given problem, the exponential function can be given by: y = P_0 * e^(-0.05t) where P_0 is the initial amount of chlorine and t is the number of hours. This equation will give the amount of chlorine remaining after t hours. The formula for exponential decay is given by:A(t) = A₀ * e^(-kt) Where A(t) is the amount after time t, A₀ is the initial amount, k is the rate of decay, and t is the time elapsed. To calculate the remaining amount of chlorine after a certain number of hours, we'll utilize the formula above .Let's start with the given information: An exponential function that gives the amount of chlorine remaining after t hours is to be found. P₀ = the initial amount of chlorine, which is 10.We can also figure out the rate of decay from the given information. The amount of chlorine is reduced by 5% per hour, which means the rate of decay is 0.05. We need to put a negative sign in front of the rate of decay to indicate that it is decreasing. Let's use the formula to find the amount of chlorine remaining after 5 hours. Then, substitute A₀ = 10, k = -0.05, and t = 5 into the equation above to obtain A(5) = 10 * e^(-0.05*5) = 7.776.So, the amount of chlorine remaining after 5 hours is 7.776.
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two triangle are similar. rye length of one side of one triangle is 4 times that of the corresponding side of the other determine the ratio of the perimeter the ratio of the areas of the polygon
The ratios for the similar triangles are given as follows:
The ratio of the perimeter is of 4.The ratio of the area is of 16.What are similar triangles?Similar triangles are triangles that share these two features listed as follows:
Congruent angle measures, as both triangles have the same angle measures.Proportional side lengths, which helps us find the missing side lengths.The length of one side of one triangle is 4 times that of the corresponding side of the other, hence the constant of the proportional relationship for the side lengths is given as follows:
k = 4.
The perimeter is measured in units, just like the side lengths, while the area is measured in units squared, hence the ratios are given as follows:
The ratio of the perimeter is of 4.The ratio of the area is of 4² = 16.More can be learned about similar triangles at brainly.com/question/14285697
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I am really confused, can anyone help?
Answer:
a = 24
b = 10
c = 26
======================================================
Explanation:
We're given a list of possible b values and they are:
4, 5, 8, 10, 12, 24
Let's make a table to show what each value of 'a' would be based on those b values above.
[tex]\begin{array}{|c|c|} \cline{1-2}b & a\\\cline{1-2}4 & 12\\\cline{1-2}5 & 14\\\cline{1-2}8 & 20\\\cline{1-2}10 & 24\\\cline{1-2}12 & 28\\\cline{1-2}24 & 52\\\cline{1-2}\end{array}[/tex]
Example calculation: If b = 4, then a = 2b+4 = 2*4+4 = 12 (first row)
I recommend using spreadsheet software to quickly compute these values. Also, a spreadsheet is useful to organize the data into a table.
Next we'll add a third column c.
This column will be computed using the formula [tex]c = \sqrt{a^2+b^2}[/tex] which is based from the pythagorean theorem [tex]a^2+b^2 = c^2[/tex]
So,
[tex]\begin{array}{|c|c|c|} \cline{1-3} & & \\b & a & c = \sqrt{a^2+b^2}\\\cline{1-3}4 & 12 & 12.6491\\\cline{1-3}5 & 14 & 14.8661\\\cline{1-3}8 & 20 & 21.5407\\\cline{1-3}10 & 24 & 26\\\cline{1-3}12 & 28 & 30.4631\\\cline{1-3}24 & 52 & 57.2713\\\cline{1-3}\end{array}[/tex]
Each decimal value mentioned is approximate. The only time c is an integer is when a = 24 and b = 10.
So that's how I got a = 24, b = 10, c = 26 as the final answer.
Estimate the sum ofand. Write an
equation.
Answer:
I believe this what you're asking:
To estimate the sum of 3.456 and 8.79, we can round each number to one decimal place, and then add them:
3.456 ≈ 3.5
8.79 ≈ 8.8
So, 3.456 + 8.79 ≈ 3.5 + 8.8 = 12.3
Therefore, we can estimate that the sum of 3.456 and 8.79 is approximately 12.3.
In equation form, this can be written as:
3.456 + 8.79 ≈ 3.5 + 8.8 = 12.3
Hope this helps! I'm sorry if it doesn't. If you need more help, ask me! :]
What is the m∠AHE
m
∠
A
H
E
to the nearest whole number?
Answer:
Below
Step-by-step explanation:
As I posted ....looks to be 147°
Please help and show work thank youuu
Based on the information provided, the total Diane will need to pay is $11.6.
How much does Diane need to pay?Let's start by calculating the total from the items she ordered, the process is shown below:
Garden salad: $2.95Monte Cristo: $7.85Soda: $1.25Total: 2.95 + 7.85 + 1.25 = $12.05Now, let´s apply the coupon by using the 10% as a reference:
12.05 x 0.9 (she will paid only 90%) = $10.845
Finally, let's calculate the tax by using its percentage and let's add it:
$10.845 x 0.0775 = $0.84
$10.845 + $0.84 = $11.6
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A: What was the percent markup on this item?
B: What was your total profit (in dollars)?
let's move like the crab, backwards.
B)
profit is simply the surplus amount or 3.75 - 2.50 = 1.25.
A)
since the item was bought for $2.50, that's our origin amount, and thus that's our 100%, what's 1.25 off of it in percentage?
[tex]\begin{array}{ccll} Amount&\%\\ \cline{1-2} 2.50 & 100\\ 1.25& x \end{array} \implies \cfrac{2.50}{1.25}~~=~~\cfrac{100}{x} \\\\\\ 2.5x=125\implies x=\cfrac{125}{2.5}\implies x=50[/tex]
Triangle ABC with vertices at A(3, 2), B(2, −1), C(−2, 1) is dilated using a scale factor of 3.5 to create triangle A′B′C′. Determine the vertex of point C′.
C′(−7, 1)
C′(−7, 3.5)
C′(−2, 3.5)
C′(7, −3.5)
I need this now I am doing the exam now there is no time to waste
According to the given information, the coordinates of C′ are (-7, 3.5).
What is the scaling of a triangle?
Scaling is a type of transformation in which the size of an object, in this case, a triangle, is changed while its shape and orientation are maintained. In other words, scaling involves multiplying the coordinates of the vertices of the triangle by a factor called the scale factor.
To dilate a triangle using a scale factor of 3.5, we need to multiply the distance between each vertex and the center of dilation by 3.5. The center of dilation can be any point, but for simplicity, we can choose the origin (0,0).
So, the coordinates of the new vertex C′ can be found by multiplying the distance between the old vertex C(-2,1) and the origin (0,0) by 3.5, and then adding this to the coordinates of the origin.
The distance between C and the origin is [tex]\sqrt{(-2-0)^2+(1-0)^2}[/tex] = [tex]\sqrt{5}[/tex] so the distance between C′ and the origin is [tex]3.5*\sqrt{5}[/tex].
Therefore, the coordinates of C′ are (-23.5, 13.5) = (-7, 3.5).
So, the answer is C′(-7, 3.5).
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a physical fitness association is including the mile run in its secondary-school fitness test. the time for this event for boys in secondary school is known to possess a normal distribution with a mean of 450 seconds and a standard deviation of 40 seconds. find the probability that a randomly selected boy in secondary school can run the mile in less than 358 seconds.
The probability that a randomly selected boy in secondary school can run the mile in less than 358 seconds is 0.0107 or 1.07%.
The probability that a randomly selected boy in secondary school can run the mile in less than 358 seconds is 0.0013.
Given data:
Mean (μ) = 450 seconds
Standard deviation (σ) = 40 seconds
We are required to find the probability that a randomly selected boy in secondary school can run the mile in less than 358 seconds.i.e., we need to find P(x < 358)
Let us first calculate the z-score.
z = (x - μ) / σ
Where,x = 358 seconds
μ = 450 seconds
σ = 40 seconds
z = (358 - 450) / 40 z = -2.3
Using a z-table or calculator, we can find the probability that corresponds to the z-score of -2.3P(z < -2.3) = 0.0107
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The mapping diagram represents a relation where x represents the independent variable and y represents the dependent variable. A mapping diagram with one circle labeled x values containing values negative 3, negative 1, 1, 3, and 5 and another circle labeled y values containing values 0, 2, and 5 and arrows from negative 3 to 0, negative 1 to 2, 1 to 0, 3 to 2, and 5 to 5. Is the relation a function? Explain. No, because for each input there is not exactly one output No, because for each output there is not exactly one input Yes, because for each input there is exactly one output Yes, because for each output there is exactly one input
The relatiοnship in the given diagram is nοt a functiοn, because fοr each input there is nοt exactly οne οutput. Sο Optiοn A is cοrrect
What are functiοns?Functiοn is a relatiοn between a set οf inputs and a set οf οutputs which are permissible. In a functiοn, fοr particular values οf x we will get οnly a single image in y. It is denοted by f(x).
Vertical line test:-
Whenever we want tο check whether a given expressiοn is a functiοn οr nοt we can apply a vertical line test, in this test we check fοr a single image οf x , we are getting a single image οr mοre.
If we get mοre images then it will nοt be a functiοn.
Fοr example, let us take, y² = 4ax
y = ±√4ax
Fοr single value οf x we get twο values οf y
Hence, it will nοt be a functiοn.
Given that,
Values οf x and values οf y
In given diagram,
fοr x = 0,
there are twο values οf y, -4 and -2
but accοrding tο definitiοn οf y, it shοuld give οnly οne value
Hence, it is nοt a functiοn
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i need help on this question
Solving a system of equations we can see that the correct option is A.
How many snacks are in each pack?Here we can define the variables:
g = snacks in one pack of granola bars.
f = snacks in one pack of fruit rolls.
And we can write a system of equations like:
10g + 6f = 152
7g + 12f = 200
If we take the difference between two times the first equation and the second, we will get:
2*(10g + 6f) - (7g + 12f) = 2*152 - 200
13g = 104
g = 104/13 = 8
There are 8 snacks in a pack of granola, and with that value we can find f.
10g + 6f = 152
6f = 152 -10g
f = (152 - 10*8)/6 = 12
We conclude that the correct option is A.
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A tamily has 4 children. Assume that each child is as likely to be a boy as it is to be a girl. Find the probability that the family has 4 girls if it is known the family has at least one giri. Answer
Therefore , the solution of the given problem of probability comes out to be the likelihood that they have four girls is 1/15.
What precisely is probability?The assessment of the probability that a claim is true or that a particular event will happen is the main objective of the constructions inside style known as criteria. Chance can be symbolised by any number between zero and 1, for which 1 typically denotes certainty range and 0 typically denotes possibility. A probability diagram illustrates the likelihood that a particular occurrence will take place. Decimal digits 0, 1, rationals with just 0% and roughly 100%.
Here,
Given that the family has at least one girl, we can use Bayes' theorem to determine the likelihood that they have four girls:
=> P(4 girls | at least one girl) = P(4 girls)
=> Times P(at least one girl | 4 girls) / P. (at least one girl)
Additionally, we are aware that the probability of having at least one female is
=> P(at least one girl = 1 - P(no girls) = 1 - (1/2)4 = 15/16.
The likelihood of not having any females is equal to the likelihood of having four boys,
which is
=> (1/2)4 = 1/16.
Given that all four of the children are female, the likelihood of producing girls is one.
Combining everything, we have:
=> P(4 females | a girl)
=> 1 * (1/16) / (15/16) = 1/15
Given that the household has at least one girl, the likelihood that they have four girls is 1/15.
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Give one pair of supplementary angles and one pair of vertical angles shown in the figure below 
Answer:
a. 6 and 2
b. 3 and 8
Step-by-step explanation: Supplementary angles add up to 180 degrees. In this figure angles 6,2 lie on the same line and a straight line has an angle measure of 180 degrees. Vertical angles are opposite to each other and have the same value. 8 and 3 are one example and on that same area, 7 and 4 are too.
Cari knows that it is a 45 mile drive from her house to the airport. She also knows that it is a 45 mile drive from her house to her grandparents house in the woods. How many miles is it directly from the airport to her grandparents house in the woods? Show your work.
the direct distance between the airport and Car's grandparents' house in the woods is 63.64 miles
Define Pythagorean theoremThe Pythagorean theorem is a fundamental principle in mathematics that relates to the sides of a right triangle. It is referred that the hypotenuse's square length, which is the side that faces the right angle, is equal to the sum of the squares of the lengths of the other two sides of a right triangle.
Let d be the distance between the airport and Car's grandparents' house in the woods. Therefore, we can use the Pythagorean theorem to solve for d:
d² = 45² + 45²
d² = 2(45²)
d = sqrt(2)× 45
Therefore, the direct distance between the airport and Car's grandparents' house in the woods is approximately 63.64 miles (since sqrt(2) is approximately 1.414).
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List the factor pairs for 36x ^ 2 : Identify which pair adds to 15x:
From the listed factor pairs, there are no factor pairs for 36x² that add up to 15x.
How to Determine Factor Pairs?A factor pair is a set of two numbers that can be multiplied together to get a specific product.
The factor pairs for 36x² are:
1 x 36x²
2 x 18x²
3 x 12x²
4 x 9x²
6 x 6x²
To identify which pair adds to 15x, we need to look for the factor pair whose two terms add up to 15x. None of the above factor pairs add up to 15x. Therefore, there is no factor pair for 36x² that adds up to 15x.
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Explain how you can use this product to determine that (15)(5)=75
The product shown below in simplest form (x+5) (x-5) and we can use this product to determine that (15) (5)= 75.
First, we need to find the product shown below in simplest form:
we have = (x+5) (x-5)
we need to explain how you can use this product to determine that,
(15) (5) = 75
for us to get the solution we will let x = 10
Then (x + 5) (x - 6) will be:
(10 + 5) (10 - 5), or we can also say that: (15)(5) = 75
Also, this could be calculated as:
10(10 - 5) + 5(10 - 5) = 100 - 50 + 50 - 25 = 75
Therefore we can say that the product shown below in simplest form (x+5) (x-5) and we can use this product to determine that (15) (5)= 75
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Full question: Find the product shown below in simplest form: (x+5) (x-5) explain how you can use this product to determine that (15) (5) = 75
solve for c
3 120 10 round your answer to the nearest tenth
The required value of the third side of the triangle is x = 11.78.
How to use law of cosine?We can use the law of cosines to find the value of the third side of the triangle. The law of cosines states that for any triangle with sides a, b, and c and angle C opposite side c,
[tex]$c^2 = a^2 + b^2 - 2ab\cos(C)$[/tex]
In this case, we have sides a = 2 and b = 10 and angle C = 120 degrees. Therefore, we can plug in these values to get:
[tex]$x^2 = 3^2 + 10^2 - 2(3)(10)\cos(120^\circ)$[/tex]
Simplifying the expression inside the parentheses gives:
[tex]$\cos(120^\circ) = -\frac{1}{2}$[/tex]
Plugging this in and simplifying further gives:
[tex]$x^2 = 9 + 100 + 30 = 139$[/tex]
Taking the square root of both sides gives:
[tex]$x = \sqrt{144} = 12$[/tex][tex]$x = \sqrt{139} = 11.78$[/tex]
Therefore, the value of the third side of the triangle is x = 11.78.
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The mean SAT score in mathematics, μ, is 53. The standard deviation of these scores is 41. A special preparation course claims that its graduates will score higher, on average, than the mean score 503. A random sample of 70 students completed the course, and their mean SAT score in mathematics was 505. At the 0. 05 level of significance, can we conclude that the preparation course does what it claims? Assume that the standard deviation of the scores of course graduates is also 41.
A. ) Null Hypothesis: B. ) Alternative Hypothesis:C. ) The Value of the test statistic:D. ) The P-Value:
A. [tex]\mu[/tex] ≤ 503
B. [tex]\mu[/tex] > 503
C. t = (505 - 503) / (41 / [tex]\sqrt{70}[/tex]) = 1.38
D. The P-value (0.086) is greater than the level of significance (0.05).
A. Null Hypothesis: The mean SAT score of the students who completed the preparation course is not significantly higher than the mean score of 503.
[tex]\mu[/tex] ≤ 503
B. Alternative Hypothesis: The mean SAT score of the students who completed the preparation course is significantly higher than the mean score of 503.
[tex]\mu[/tex] > 503
C. The Value of the test statistic:
We apply the algorithm below to determine the test statistic:
[tex]t = (\bar x - \mu) / (s / \sqrt{n} )[/tex]
When s is the sample standard deviation, n is the sample size, and x is the sample mean and is the predicted population mean.
In this case,
[tex]\bar x[/tex] = 505,
[tex]\mu[/tex] = 503,
s = 41,
n = 70.
t = (505 - 503) / (41 / [tex]\sqrt{70}[/tex]) = 1.38
D. The P-Value:
We want to test whether the mean SAT score of the students who completed the preparation course is significantly higher than the mean score of 503 at the 0.05 level of significance.
Using a t-distribution table with 69 degrees of freedom (df = n-1), we find that the area to the right of 1.38 is 0.086.
Since this is a one-tailed test (we are testing for[tex]\mu[/tex] > 503), the P-value is 0.086.
Since the P-value (0.086) is greater than the level of significance (0.05), we fail to reject the null hypothesis. Therefore, we do not have sufficient evidence to conclude that the preparation course does what it claims, i.e., the mean SAT score of the students who completed the preparation course is not significantly higher than the mean score of 503.
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A right triangle has a rise of 16 and a run of 4. A similar right triangle with a run of 5 will have a rise of?
Answer:
It will have a rise of 20.
Step-by-step explanation:
We can use ratios:
Rise : Run = 16 : 4 = 4 : 1 = 20 : 5
Hope this helps!
HELP ME!!!! THIS IS DUE TODAY
Answer:
No, it is not reasonable. The scaled drawing would be 12 inches wide by 30 inches tall.
Step-by-step explanation:
First, scale down the sides. 48in divided by 4 is 12in , and 120 in divided by 4 is 30 in. So no, it would not fit on her paper.
Instructions: Rewrite the equation in Standard Form. y+3=−(x−5)
Show that there is no finite triangle in hyperbolic geometry that achieves the maximum area bound.
As a result, the total area of the four triangles (PTA, PTB, PTC, and ABC) equation exceeds the size of T, which contradicts the premise that T has the greatest area.
What is equation?A math equation is a technique that links two assertions and denotes equivalence using the equals sign (=). In algebra, an equation is a mathematical statement that proves the equality of two mathematical expressions. For example, in the equation 3x + 5 = 14, the equal sign separates the numbers 3x + 5 and 14. A mathematical formula may be used to understand the link between the two phrases written on either side of a letter. The logo and the programmed are usually interchangeable. As an example, 2x - 4 equals 2.
The area of a triangle in hyperbolic geometry is determined by the formula:
Area = K ∙ (α + β + γ - π)
where K is a constant that depends on the curvature of the space and,, and are the triangle's angles.
we want to emphasise the expression + + -.
Consider the following two situations:
Case 1: T has angles that add up to be less than.
Moreover, the total of the areas of triangles PTA, PTB, and PTC is larger than the area of T since their angles add up to more than T's angles. As a result, the total area of the four triangles (PTA, PTB, PTC, and ABC) exceeds the size of T, which contradicts the premise that T has the greatest area.
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a. Solve - 11y -13>42. Graph the solution on a number line.
The solution to the inequality - 11y -13>42 is y < -5.
What is inequality?An inequality is a mathematical statement that compares two quantities using symbols such as <, >, ≤, or ≥ to indicate whether one is less than, greater than, less than or equal to, or greater or equal to the other.
To solve the inequality -11y - 13 > 42, we need to isolate y on one side of the inequality.
-11y - 13 > 42
First, we can add 13 to both sides:
-11y > 55
Next, we can divide both sides by -11, remembering to reverse the direction of the inequality because we are dividing by a negative number:
y < -5
To graph this on a number line,
Here is a rough sketch of the number line:
<=====(●)------------------------
-5
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if (5 x - 7) is a factor of the expression: 5 x ^2 - 2 x - 7 then the other factor is
Answer:
(x + 3)
Step-by-step explanation:
Multiply out and simplify (x-8)²
(x-8)²
(x-8)(x-8)
x(x-8) - 8 (x-8)
x² -8x -8x + 64
x²-16x+64
Lupita deposited $5,000 into a savings account that earns 2% interest compounded annually. Rounded to the nearest cent, how much interest will Lupita have earned after 48 months?
The difference between the total and the initial contribution, which is: Interest [tex]= $5,409.09 - $5,000, I[/tex] nterest $409.09, is the amount of interest earned. Lupita will have accrued interest of $409.09, rounded to the closest penny, after 48 months.
We can use the compound interest calculation to determine how much money Lupita will receive after 48 months:
Where: [tex]A = P(1 + r/n)(nt)[/tex] A represents the whole sum after t years.
P represents the primary sum.
The yearly interest rate is r.
Given that interest is compounded yearly, [tex]P = $5,000, r = 0.02, n = 1,[/tex]and[tex]t = 48/12 = 4[/tex] (since the time is given in months, we need to convert to years by dividing by 12).
Hence, the equation is: [tex]A = $5,000(1 + 0.02/1)^(1*4)[/tex]
[tex]A = $5,000(1.02)^4\sA ≈ $5,409.09[/tex]
The amount that differs between the total and the original deposit that represents interest earned is: Interest equals [tex]$5,000 - $5,409.09$409.09[/tex] in interest
Lupita will have accrued interest of $409.09, rounded to the closest penny, after 48 months.
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Q.2) A jar contains three balls numbered 1,2, and 3. If two balls are drawn: a) Write the probability space? b) what is the probability that the sum of the numbers is 4 ? c) what is the probability that the sum of the numbers is at least 4 ?
a) The probability space is (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3).
b) The probability that the sum of the numbers drawn is 4 is 2/9.
c) The probability that the sum of the numbers is at least 4 is 2/3.
a) The probability of an event is measured between 0 and 1, and the probability space is a collection of all probable results, hence the probability space is:
P= (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)
Hence, the probability space can be written as a set, S = {1,2,3}.
b) If 2 balls are drawn and sum of the numbers is 4, then there could be only one probable way of drawing the balls, which is ball 1 and 3, so probability that the sum of the numbers is 4 is:
P = 2/9
c) If 2 balls are drawn and sum of the numbers is at least 4, then probable ways of drawing the balls are:
(1,3), (2,2), (2,3), (3, 1), (3, 2), and (3, 3)
Hence, the probability that the sum of the numbers is at least 4 = 6/9 = 2/3
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Raven usually does 5 out of every 7 math problems correctly. Which table accurately shows how many math problems she will do correctly out of 14, 21, or 28 total problems?
Answer:
B.
Step-by-step explanation:
The proportions should be exactly the same for each set of questions and correct questions.
Based on Raven's history, she get 5 of 7.
Choice B. shows exactly that with a scalar of 2, 3, and 4. For Row 2, the set is twice as large, so Raven gets exactly twice as many questions correct and exactly twice as many incorrect.
For Option B., the [tex]\frac{correct}{total}[/tex] always simplifies to [tex]\frac{5}{7}[/tex] for every row.
If the sum of the 2nd and 7th term of an arithmetic progression is 25 and the fifth term is 15 find the common difference
Answer: Let's call the first term of the arithmetic progression "a" and the common difference "d". Then we can use the following formulas to find the second, fifth, and seventh terms:
- The second term is a + d.
- The fifth term is a + 4d.
- The seventh term is a + 6d.
We are given that the sum of the second and seventh terms is 25:
(a + d) + (a + 6d) = 25
Simplifying this equation, we get:
2a + 7d = 25 ...(1)
We are also given that the fifth term is 15:
a + 4d = 15 ...(2)
Now we can solve for the common difference "d" by using equations (1) and (2) to form a system of two equations in two variables. One way to do this is to solve equation (2) for "a" in terms of "d", and then substitute that expression for "a" into equation (1):
a = 15 - 4d ...(3)
Substituting equation (3) into equation (1), we get:
2(15 - 4d) + 7d = 25
Simplifying this equation, we get:
30 - 8d + 7d = 25
Solving for "d", we get:
d = -5
Therefore, the common difference is -5.
Step-by-step explanation:
gina prepara un postre para 8 personas usa 1/2 de libra de mantequilla 1/4 de libra de azucar ,una lib a de harina y 3/2 libra de queso cuantas libras de ingredientes necesita si para preparar la receta para 16 personas cuantas libras necesita
The amounts needed for the dessert for 16 people is given as follows:
Butter: 1 lb.Sugar: 0.5 lb.Flour: 2 lb.Cheese: 3 lb.How to obtain the amounts?The amounts are obtained applying the proportions in the context of the problem.
For 8 people, the amounts of the ingredients are given as follows:
Butter: 0.5 lb.Sugar: 0.25 lb.Flour: 1 lb.Cheese: 1.5 lb.With 16 people, the number of people doubles, hence the amount of ingredients also doubles, thus the needed amounts are given as follows:
Butter: 1 lb.Sugar: 0.5 lb.Flour: 2 lb.Cheese: 3 lb.TranslationGina is preparing a recipe for 8 people, and the amounts are given as follows:
Butter: 0.5 lb.Sugar: 0.25 lb.Flour: 1 lb.Cheese: 1.5 lb.The problem asks for the necessary amounts for 16 people.
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Next year, Frank plans to complete a triathlon. It will consist of a 2. 25-km swim, a 65-km bike ride, and a 20-km run. With which set of rates could Frank complete each event to finish the triathlon within 7. 5 hours?
The set o rates that Frank could use to finish the events within 7.5 hours is (0.75 km/h, 25 km/h, 12.5 km/h).
What is the rates needed by Frank to complete the events?
To determine the rates at which Frank must complete each event to finish the triathlon within 7.5 hours, we can use the formula:
time = distance / rate
where;
time is the amount of time it takes to complete the event, distance is the distance of the event, and rate is the speed at which Frank completes the event.Let x be the rate at which Frank completes the swim,
y be the rate at which he completes the bike ride, and
z be the rate at which he completes the run.
Then, we have the following three equations:
2.25 / x + 65 / y + 20 / z = 7.5
x > 0, y > 0, z > 0
We want to find the set of rates (x, y, z) that satisfy these equations and inequalities.
Here is one possible set of rates that would allow Frank to complete the triathlon within 7.5 hours:
x = 0.5 km/hour
y = 20 km/hour
z = 10 km/hour
Using these rates, we can calculate the time it would take Frank to complete each event:
Swim: 2.25 km / 0.5 km/hour = 4.5 hours
Bike ride: 65 km / 20 km/hour = 3.25 hours
Run: 20 km / 10 km/hour = 2 hours
The total time would be:
4.5 + 3.25 + 2 = 9.75 hours
This is greater than 7.5 hours, so Frank would need to increase his rates to finish within the time limit.
Here are some possible sets of rates that would allow him to do so:
x = 0.75 km/hour, y = 25 km/hour, z = 12.5 km/hour
x = 1 km/hour, y = 30 km/hour, z = 15 km/hour
x = 1.25 km/hour, y = 35 km/hour, z = 17.5 km/hour
Using any of these sets of rates, Frank would be able to complete the triathlon within 7.5 hours.
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