Pat receives a series of four annual federally subsidized student loans, each for $5600 at 6.9%. To defray rising costs for her senior year, 3 years after acquiring
the first loan she takes out a private student loan for $3900 at 7.3% interest with a term of 10 years and capitalizes the Interest for her last year of college. She
graduates 9 months after getting the private loan. Payments on all loans are deferred until 6 months after graduation. Find her monthly payment.

The solution of the given differential equation that satisfies the initial condition y(0) = 0 is 0 = (5/2)(0)² - (1/4)(ln(0))^2 - (5/3)(0)³ + C.

To find the solution of the given differential equation that satisfies the initial condition y(0) = 0, we will follow these steps,

1. Identify the differential equation: dy/dx = 5x - (ln(x)/2) - 5x²
2. Integrate both sides of the equation with respect to x.

Integral of dy = Integral of (5x - (ln(x)/2) - 5x²) dx

Since y(0) = 0, we have:

y(x) = Integral of (5x - (ln(x)/2) - 5x²) dx

3. Perform the integration:

y(x) = (5/2)x² - (1/4)(ln(x))^2 - (5/3)x³ + C

4. Determine the value of the constant C using the initial condition y(0) = 0:

0 = (5/2)(0)² - (1/4)(ln(0))^2 - (5/3)(0)³ + C

Since ln(0) is undefined, we cannot solve for C using the initial condition y(0) = 0. However, the given initial condition is not consistent with the differential equation, so there may be an error in the problem statement.

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a. P(x < 79) = 1 - P(x > 79) = 1 - 0.17 = 0.83. c. P(x = 79) For a continuous random variable, the probability of x taking any specific value (like x = 79) is always 0, because the probability is spread across an infinite number of possible values within the range.

a. To find P(x < 79), we can use the complement rule: P(x < 79) = 1 - P(x > 79). We are given that P(x > 79) = 0.17, so:

P(x < 79) = 1 - 0.17 = 0.83

Therefore, the probability that x is less than 79 is 0.83.

b. To find P(x < 29), we can use the fact that the probability distribution for a continuous random variable is continuous and smooth, which means that P(x < 29) = 0.

This is because the interval [30, 79] already has a probability of 0.26, so there can be no additional probability assigned to values less than 30.

Therefore, the probability that x is less than 29 is 0.

c. To find P(x = 79), we can use the fact that the probability of a specific value for a continuous random variable is 0.

This is because the probability distribution is continuous and smooth, so the probability of any specific value is infinitely small.

Therefore, the probability that x is equal to 79 is 0.

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The value of x is 13 and can be calculated by setting the number of students who played soccer and rugby (S ∩ R) but not Gaelic football equal to x - 4, and then solving for x.

We know that:

65 students played Gaelic football (G)

57 students played soccer (S)

34 students played rugby (R)

42 students played Gaelic football and soccer (G ∩ S)

16 students played Gaelic football and rugby (G ∩ R)

x students played soccer and rugby (S ∩ R)

4 students played all three sports (G ∩ S ∩ R)

6 students played none of the sports listed

To fill in the Venn diagram, we can start with the three circles representing Gaelic football (G), soccer (S), and rugby (R), and add the numbers in each region based on the information provided. Let's go region by region:

The region inside all three circles (G ∩ S ∩ R) has 4 students.

The region inside both Gaelic football and soccer circles (G ∩ S) but outside the rugby circle has 42 - 4 - 16 = 22 students.

The region inside both Gaelic football and rugby circles (G ∩ R) but outside the soccer circle has 16 - 4 = 12 students.

The region inside both soccer and rugby circles (S ∩ R) but outside the Gaelic football circle has x - 4 = x - 4 students.

The region inside only the Gaelic football circle (G) but outside the other two circles has 65 - 4 - 22 - 16 - 6 = 17 students.

The region inside only the soccer circle (S) but outside the other two circles has 57 - 4 - 22 - x + 4 - 6 = 25 - x students.

The region inside only the rugby circle (R) but outside the other two circles has 34 - 4 - 16 - x + 4 - 6 = 8 - x students.

The region outside all three circles has 6 students.

Total number of students who played soccer = S + (S ∩ R) + (G ∩ S

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The population mean is equal to 80 for the mean of a sampling distribution of sample means with n=49, and the standard deviation is equal to 1/√(n).

The standard deviation is a statistician's gauge of a group of values' degree of dispersion or variation. While a high standard deviation suggests that the values are dispersed over a wider range, a low standard deviation suggests that the values tend to be close to the mean (also known as the expected value) of the set.With a sample size of 49, the mean of a sampling distribution of sample means is equal to the population mean of 801.

With a sample size of n=49, the standard deviation of a sampling distribution of sample means is identical.

n=49 is equal to σ/√(n)

= 7/√(49) = 1¹

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Answer:

Scientific Notation:

[tex]8.1*10^7[/tex]

Expanded form:

81000000

Hope this helps :)

Pls brainliest...

The value of ''(3squared)squared times (10cubed)squared'' is,

8.1 × 10⁶

We have,

Expression is,

= (3squared)squared times (10cubed)squared

It can be written as,

(3squared)squared times (10cubed)squared

(3²)² × (10³)²

9² × 1000²

81 × 1000000

8,10,00,000

8.1 × 10⁶

Thus, The value of ''(3squared)squared times (10cubed)squared'' is,

8.1 × 10⁶

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Answer:

[tex]2[/tex] × [tex]5^{2}[/tex]

Step-by-step explanation:

2 X 5 X 5 = 50

Hope this helps

Answer:

To factor the quadratic function g(x) = 8x^2 - 2x - 3, we can use the following steps:

Step 1: Multiply the coefficient of the x^2 term (8) and the constant term (-3).

8 * -3 = -24

Step 2: Find two numbers that multiply to give the result from step 1 (-24) and add up to the coefficient of the x term (-2).

The two numbers that meet these criteria are -6 and +4, since -6 * 4 = -24 and -6 + 4 = -2.

Step 3: Rewrite the middle term (-2x) using the two numbers found in step 2 (-6 and +4).

8x^2 - 6x + 4x - 3

Step 4: Group the terms and factor by grouping.

2x(4x - 3) + 1(4x - 3)

Step 5: Factor out the common binomial (4x - 3).

(4x - 3)(2x + 1)

So, the factored form of the quadratic function g(x) = 8x^2 - 2x - 3 is (4x - 3)(2x + 1).

The mean of all the sample means obtained from all random samples of a certain sample size in a sampling distribution is an example of a biased statistic is False.

The mean of all the sample means obtained from all random samples of a certain sample size in a sampling distribution is an example of an unbiased statistic. This is because the sampling distribution of the mean is centered at the true population mean, and the mean of all sample means provides an estimate of that true population mean without any systematic over- or under-estimation.

However, individual sample means can be biased if there are any issues with the sampling process or if the sample is not representative of the population.

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Here are the step-by-step workings:

Circumference = π * Diameter

Circumference = 3.14 * 50 units

Circumference = 157 units

Rounded to the nearest hundredth:

Circumference = 157.00 units

Answer:

ask your teacher

What is the circumference of a circle with a diameter of 50 units? Use π = 3.14 and round your answer to the nearest hundredth

The given series is conditionally convergent.

We can use the alternating series test to show that the series converges. First, we can rewrite the terms of the series as:

The terms of the series are decreasing in absolute value and approach zero as n approaches infinity. Also, the series is alternating in sign, so we can apply the alternating series test. Therefore, the series converges.

To determine whether the series is absolutely convergent or conditionally convergent, we need to check the convergence of the series of absolute values:

We can use the limit comparison test to compare this series with the series ∑ (1/√(n)). We have:

lim (n/√(n³ + 5)) / (1/√(n)) = lim (n*√(n)) / √(n³ + 5) = lim 1 / √(1 + 5/n²) = 1

Since this limit is a positive finite number, the series ∑ |an| and the series ∑ (1/√(n)) have the same behavior. The series ∑ (1/√(n)) is a p-series with p=1/2, which is known to be divergent. Therefore, the series ∑ |an| is also divergent. Since the original series is convergent but |an| is divergent, the original series is conditionally convergent.

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answer in attached image

(x, y) → (-y, x)

We would need a sample size of 16 to construct a 95% confidence interval to estimate the average air travel cost for a college student with a margin of error of $50. The critical value for a 95% confidence interval is approximately 1.96.

To determine the sample size needed to construct a 95% confidence interval to estimate the average air travel cost for a college student with a margin of error of $50, we need to use the formula:

n = (zα/2 * σ / E)^2

where:
- n is the sample size
- zα/2 is the critical value for the desired confidence level, which is 1.96 for 95% confidence interval
- σ is the standard deviation of the population, which is unknown, so we use the sample standard deviation as an estimate
- E is the margin of error, which is $50

Assuming that we have a pilot sample of air travel costs for college students, we can use the sample standard deviation as an estimate for the population standard deviation.

Let's say the sample standard deviation is $200.

Plugging in the values, we get:

n = (1.96 * 200 / 50)^2
n = 15.36

Since we can't have a fraction of a sample, we need to round up to the nearest whole number, which gives us a sample size of 16.

To calculate the required sample size for a 95% confidence interval with a margin of error of $50, we need some information about the population standard deviation (σ) and the critical value (Z) associated with the desired confidence level.

Since the problem does not provide the population standard deviation, I'll assume it is known or estimated from a previous study.

Let's call it σ.The margin of error (E) formula for a confidence interval is:

E = Z * (σ / √n)

Where:
E = margin of error ($50)
Z = critical value (1.96 for a 95% confidence interval)
σ = population standard deviation
n = sample size

We need to solve for n:

50 = 1.96 * (σ / √n)

To isolate n, we can follow these steps:

1. Divide both sides by 1.96:
50 / 1.96 = σ / √n

2. Square both sides:
(50 / 1.96)^2 = (σ^2 / n)

3. Multiply both sides by n:
(50 / 1.96)^2 * n = σ^2

4. Divide both sides by (50 / 1.96)^2:
n = σ^2 / (50 / 1.96)^2

Now, plug in the known or estimated value for σ, and calculate the required sample size (n). Remember to round up to the nearest whole number, as you cannot have a fraction of a sample.

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Therefore, () r(t) = 4 - 8t evaluated at [tex]t=0[/tex] is 4, at [tex]t=1[/tex] is -4, and at [tex]t=4[/tex]is -28.

To evaluate the dot product ()r(t), we first need to find the coordinates of the vector :

() = ⟨2, -2, 4⟩

Then we can substitute the coordinates of r(t) into the dot product formula:

[tex]()r(t) = (2t^2 - 2 - 2t^2, -2t^2 - 2t^3, 4 - 6t) ⋅ ⟨2, -2, 4⟩[/tex]

Simplifying this expression yields:

[tex]()r(t) = 4 - 8t[/tex]

To evaluate () r(t) at different values of t, we substitute those values into the expression we just derived:

[tex]() r(0) = 4 - 8(0) = 4[/tex]

[tex]() r(1) = 4 - 8(1) = -4[/tex]

[tex]() r(4) = 4 - 8(4) = -28[/tex]

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The probability that a student attends a live class at least 2 days a week is 0.58 and the probability that a student attends a live class less than 2 days a week is 0.56.

To find the probability that a student attends a live class at least 2 days a week, we need to add up the probabilities of attending class for 2, 3, 4, and 5 days. This is because attending 0 or 1 day a week means attending less than 2 days, so we need to exclude those probabilities.
P(attending at least 2 days) = P(x = 2) + P(x = 3) + P(x = 4) + P(x = 5)
= 0.22 + 0.22 + 0.10 + 0.04
= 0.58
Therefore, the probability that a student attends a live class at least 2 days a week is 0.58.
To find the probability that a student attends a live class less than 2 days a week, we need to add up the probabilities of attending 0 or 1 day a week.
P(attending less than 2 days) = P(x = 0) + P(x = 1)
= 0.34 + 0.22
= 0.56
Therefore, the probability that a student attends a live class less than 2 days a week is 0.56.

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The power series of the function 3/((x-2)(x+1)) is:

-3/4 - 1/4(x-2) + 1/4(x-2)² - 1/4(x-2)³- 1/2 - 1/2(x+1) - 1/2(x+1)² - 1/2(x+1)³ + ...

To find the power series of the function 3/((x-2)(x+1)), we first need to find the partial fraction decomposition of the function:

3/((x-2)(x+1)) = A/(x-2) + B/(x+1)

To solve for A and B, we need to find a common denominator on the right-hand side:

3 = A(x+1) + B(x-2)

Setting x = 2, we get:

3 = A(3)

A = 1

Setting x = -1, we get:

3 = B(-3)

B = -1

Therefore, we have:

3/((x-2)(x+1)) = 1/(x-2) - 1/(x+1)

Now we can use the formula for the geometric series:

1/(1 - t) = 1 + t + t²+ t³ + ...

to write the power series of each term in the partial fraction decomposition. Substituting t = x-2 for the first term and t = -x-1 for the second term, we get:

1/(x-2) = -1/4 - 1/4(x-2) + 1/4(x-2)² - 1/4(x-2)³ + ...

1/(x+1) = -1/2 - 1/2(x+1) - 1/2(x+1)² - 1/2(x+1)³ - ...

Combining the two series, we have:

3/((x-2)(x+1)) = -3/4 - 1/4(x-2) + 1/4(x-2)² - 1/4(x-2)³ - 1/2 - 1/2(x+1) - 1/2(x+1)² - 1/2(x+1)³ + ...

Therefore, the power series of the function 3/((x-2)(x+1)) is:

-3/4 - 1/4(x-2) + 1/4(x-2)² - 1/4(x-2)³- 1/2 - 1/2(x+1) - 1/2(x+1)² - 1/2(x+1)³ + ...

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The statement " a sort of o(nlogn) is always preferable to a sort of o(n 2)" is true because sorting algorithms with O(n log n) time complexity have a lower rate of growth and are generally more efficient than sorting algorithms with O(n^2) time complexity

In general, it is true that a sorting algorithm with a time complexity of O(n log n) is preferable to a sorting algorithm with a time complexity of O(n^2), assuming other factors such as memory usage and stability are comparable.

This is because the time complexity of an algorithm describes the rate at which the algorithm's running time increases as the input size grows. In the case of sorting, O(n log n) algorithms, such as merge sort or quicksort, have a much lower rate of growth than O(n^2) algorithms, such as bubble sort or insertion sort.

This means that as the input size grows larger, the time required to sort the input using an O(n^2) algorithm can become prohibitively long, while an O(n log n) algorithm can still be practical.

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The required answer is f(x) = x + 4 and f(x) = 2x - 5

The group of answer choices, both f(x) = x + 4 and f(x) = 2x - 5 have well-defined inverses as they are both one-to-one and onto functions.

To select the function that has a well-defined inverse from the group of answer choices, we need to look for the function that satisfies the horizontal line test. The horizontal line test states that a function has a well-defined inverse if no horizontal line intersects the graph of the function more than once.

The company raised a $6 million Series A funding in 2016, led by Crosslink Capital with participation from Bertelsmann Digital Media Investments.
Out of the four answer choices, the only function that satisfies the horizontal line test is f:→f(x)=|x|. Therefore, the function f:→f(x)=|x| has a well-defined inverse.

To select the function that has a well-defined inverse, we need to identify the function that is both one-to-one and onto. Here are the given functions:

1. f(x) = |x|
2. f(x) = x + 4
3. f(x) = ⌈x/2⌉
4. f(x) = 2x - 5

the inverse function of a function f (also called the inverse of f) is a function that undoes the operation of f. The inverse of f exists if and only if f is bijective, and if it exists, is denoted by, [tex]f-1[/tex]
Now let's analyze each function:

1. f(x) = |x| is not one-to-one because f(1) = f(-1) = 1.
2. f(x) = x + 4 is one-to-one and onto, as every input has a unique output and every output can be achieved by a unique input.
3. f(x) = ⌈x/2⌉ is not one-to-one because f(1) = f(2) = 1.
4. f(x) = 2x - 5 is one-to-one and onto, as every input has a unique output and every output can be achieved by a unique input.
the concept of an inverse element generalises the concepts of opposite (−x) and reciprocal (1/x) of numbers.

Among the group of answer choices, both f(x) = x + 4 and f(x) = 2x - 5 have well-defined inverses as they are both one-to-one and onto functions.

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Answer:

i believe it is D. because you need multiple sets of data. :)

The final thickness of the silicon dioxide on a wafer is given by: Initial thickness + (growth rate x oxidation time)

To calculate the final thickness of the silicon dioxide on a wafer, you will need to know the initial thickness of the oxide layer and the duration of the oxidation process.

The growth rate of silicon dioxide is dependent on temperature and can be determined from the literature. Once you have this information, you can use the following formula to calculate the final thickness:

Final thickness = initial thickness + (growth rate x oxidation time)

It is important to note that the final thickness may be affected by any post-oxidation processing steps, such as etching or cleaning, that may remove some of the oxide layer.

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(a)  left-hand critical value is 17.71, (b) the right-hand critical value is 46.98 and (c) the 90% confidence interval for the population standard deviation is: 9.58 ≤ σ ≤ 17.45.

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The option that is the  difference of cubes is option A) 125x²¹- 64y³  

125x²¹ - 64y³, can be written as the difference of cubes due to:

a³ - b³ = (a - b) (a² + ab + b²)

Hence 125x²¹ - 64y³ = (5x⁷ - 4y) (25x¹⁴ + 20x⁷y + 16y²)

Note that:

x⁶ + 27y⁹ = (x²)³ + (3y³)³  - sum of cubes

3x⁹ - 64y³ - the first term is not a cube

27x¹⁵ - 9y³ - the second term is not a cube

125x²¹- 64y³ = (5x⁷)³ - (4y)³ - difference of cubes

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To find the limit of the sequence, we need to take the value of "n" to infinity.
So, let's divide both the numerator and denominator by the highest power of "n", which is "2n^2".
an = (2n^2 + 4n + 3) / (8n^2 + 6n + 6)

Now, as "n" tends to infinity, the terms with lower powers of "n" become insignificant. Therefore, we can neglect the terms "4n" and "6n" in the numerator and denominator.
an = (2n^2 + 3) / (8n^2 + 6n + 6)
Now, taking the limit of the sequence as "n" tends to infinity:
limit = lim(n → ∞) [(2n^2 + 3) / (8n^2 + 6n + 6)]
Using the rule of L'Hopital's rule, we can differentiate the numerator and denominator separately with respect to "n".
limit = lim(n → ∞) [(4n) / (16n + 6)]
As "n" tends to infinity, the denominator becomes very large, and the term "6" becomes insignificant. So,
limit = lim(n → ∞) [(4n) / (16n)]
limit = lim(n → ∞) [1 / 4]
limit = 1/4
Therefore, the limit of the sequence is 1/4.

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The value of c for each situation is as follows: (a) c = -0.80 (b) c = 1.64 (c) c = 2.55 (d) c = -0.50.

We will use the z-table to find the corresponding z-scores.

(a) For p(z < c) = 0.2119, look for 0.2119 in the z-table, and find the closest value.

In this case, it is approximately 0.2118, which corresponds to a z-score of -0.80.

So, c = -0.80.

(b) For p(-c < z < c) = 0.9030, first we need to find p(z < c) since it is symmetrical around the mean.

This means p(z < c) = 1 - (1 - 0.9030) / 2 = 0.9515.

Look for 0.9515 in the z-table, and the closest value is 0.9517, which corresponds to a z-score of 1.64.

So, c = 1.64.

(c) For p(z < c) = 0.9948, look for 0.9948 in the z-table, and find the closest value.

In this case, it is approximately 0.9949, which corresponds to a z-score of 2.55.

So, c = 2.55.

(d) For p(z > c) = 0.6915, we need to find p(z < c) first.

p(z < c) = 1 - 0.6915 = 0.3085.

Look for 0.3085 in the z-table, and the closest value is 0.3085, which corresponds to a z-score of -0.50.

So, c = -0.50.

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The invertible (AB)^-1 exists and is equal to B^-1A^-1. Yes, that is correct.

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The shape of the distribution of the box plot is described as: B: It is Skewed

The box plot shape is used to indicate if a statistical particular set is either normally distributed or skewed.

There is a property of the box plot i.e. When the median is in the center of the box, and the whiskers are about the same on both flanks of the box, then the distribution is symmetric. However, when the median is anywhere to the box except the center, and if the whisker is more concise on the left or right end of the box, then the distribution is skewed.

In this question, we can clearly see that  the whisker is more concise on the right end of the box, and as such we can say that the distribution is skewed.

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The order of 8 + 12Z in the factor group Z/12Z is 3.

This is because the order of an element a in a group G is the smallest positive integer n such that aⁿ = e, where e is the identity element of G. In this case, (8 + 12Z)³ = 8³ + 12Z = 8 + 12Z = 0 + 12Z, which is the identity element of Z/12Z. Therefore, the order of 8 + 12Z is 3.

To find the order of an element in a factor group, we first need to determine the cosets of the group modulo the subgroup. In this case, we have Z/12Z, which is the integers modulo 12. The subgroup is 12Z, which consists of all multiples of 12.

We can write the cosets of 12Z as {0 + 12Z, 1 + 12Z, 2 + 12Z, ..., 11 + 12Z}. Each of these cosets contains an element that is congruent to 8 modulo 12. For example, 8 + 12Z is in the coset 8 + 12Z, and 20 + 12Z is in the coset 8 + 12Z.

To find the order of 8 + 12Z, we need to find the smallest positive integer n such that (8 + 12Z)ⁿ is equal to the identity element of Z/12Z, which is 0 + 12Z. We can compute (8 + 12Z)² as (8 + 12Z)(8 + 12Z) = 64 + 96Z = 4 + 12Z, since 64 is congruent to 4 modulo 12 and 96 is a multiple of 12. Therefore, (8 + 12Z)² is not equal to the identity element.

Next, we compute (8 + 12Z)³ as (8 + 12Z)(8 + 12Z)(8 + 12Z) = 512 + 864Z = 8 + 12Z, since 512 is congruent to 8 modulo 12 and 864 is a multiple of 12. Therefore, (8 + 12Z)³ is equal to the identity element, and the order of 8 + 12Z is 3.

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The population of the country in 2003 was 979 million.

We are given that;

a=979e 0.0008t

Now,

To find the population of the country in 2003, we need to plug in t = 0 into the model, since 2003 is the starting year.

a = 979e^(0.0008t)

a = 979e^(0.0008(0))

a = 979e^0

a = 979(1)

a = 979

Therefore, by the exponential mode the answer will be 979 million.

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The population of the country in 2003 was 979 million.

We are given that;

a=979e 0.0008t

Now,

To find the population of the country in 2003, we need to plug in t = 0 into the model, since 2003 is the starting year.

a = 979e^(0.0008t)

a = 979e^(0.0008(0))

a = 979e^0

a = 979(1)

a = 979

Therefore, by the exponential mode the answer will be 979 million.

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Therefore, Melinda used 6 $0.10 stamps in the given equation.

Let's say Melinda used x $0.02 stamps and y $0.10 stamps.

From the problem, we know that:

x + y = 15 (the total number of stamps used is 15)

0.02x + 0.1y = 0.78 (the total value of the stamps used is $0.78)

To solve for y, we can use the first equation to solve for x:

x = 15 - y

Substituting into the second equation:

0.02(15 - y) + 0.1y = 0.78

Expanding and simplifying:

0.3 - 0.02y + 0.1y = 0.78

0.08y = 0.48

y = 6

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