For the linear function(straight line) y = 7x - 3,ar function(straight line) y = 7x - 3,

a) What is the slope of the given function?



b) Is the function increasing or decreasing?



c) What is the y-intercept(0, b) of the given function?

Answer: 5 1/2 (5.5)

Step-by-step explanation:

add 4 2/3 to 5/6

The critical point (1, -1) represents a relative minimum of f(x, y) with a value of -1/3 and (1, -1) is the only extremum or relative maximum of the function.

To find the relative maximum and minimum values of the function f(x, y) = ([tex]x^3[/tex])/3 + 2xy +[tex]y^2[/tex] - 3x + 1, we need to analyze its critical points and classify them using the second partial derivative test.

To find the critical points, we need to compute the partial derivatives of f with respect to x and y and set them equal to zero:

∂f/∂x = [tex]x^2[/tex] + 2y - 3 = 0

∂f/∂y = 2x + 2y = 0

Solving these equations simultaneously, we find x = 1 and y = -1.

Thus, the critical point is (1, -1).

Next, we need to compute the second partial derivatives and evaluate them at the critical point:

∂²f/∂x² = 2

∂²f/∂y² = 2

∂²f/∂x∂y = 2

Now, we can use the second partial derivative test to classify the critical point.

The discriminant D = (∂²f/∂x²) × (∂²f/∂y²) - [tex]\left(\frac{{\partial^2 f}}{{\partial x \partial y}}\right)^2[/tex] = (2)(2) - [tex](2)^2[/tex] = 0.

Since D = 0, the test is inconclusive.

To determine the nature of the critical point, we can examine the function near the critical point.

Evaluating f at the critical point (1, -1), we find f(1, -1) = [tex](1^3)[/tex]/3 + 2(1)(-1) + [tex](-1)^2[/tex] - 3(1) + 1 = -1/3.

Hence, the critical point (1, -1) represents a relative minimum of f(x, y) with a value of -1/3.

There are no other critical points to consider, so we can conclude that (1, -1) is the only extremum of the function.

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Answer:

It takes more 1/9th because they ar smaller than 1/6th. You would need three 1/9ths and two 1/6th in order for it to equal 1/3

a.0.05 ≤ P (χ²(17) <  (18 - 1)35.05/σ²M < χ²(0.95)(17))0.05 ≤ P (χ²(17) <  577.57/σ²M < χ²(0.95)(17))0.05 ≤ P (χ²(17) <  577.57/σ²M < 28.412)The above inequality represents the 90%

b. 0.05 ≤ P (χ²(7) <  (8 - 1)18.40/σ²F < χ²(0.95)(7))0.05 ≤ P (χ²(7) <  122.18/σ²F < χ²(0.95)(7))0.05 ≤ P (χ²(7) <  122.18/σ²F < 14.067)

The above inequality represents the 90% confidence interval estimate for sigma2F.

a. 90% confidence interval estimate for sigma2M:We are given that the sample variance is s²M=35.05 and a sample of 18 male students was asked how much they spent on textbooks. We are also given that the amount spent on textbooks is normally distributed for both the populations of male students.

Using the Chi-Square distribution, we have:  (n - 1)s²M/σ²M follows a Chi-Square distribution with n - 1 degrees of freedom.

Then,  (n - 1)s²M/σ²M ~ χ²(n - 1)For a 90% confidence interval estimate, we can write: 0.05 ≤ P (χ²(17) <  (n - 1)s²M/σ²M < χ²(0.95)(17))

Using the table of chi-square values with (n - 1) degrees of freedom, we have:χ²(0.05)(17) =  8.567χ²(0.95)(17) =  28.412

Substituting the values, we have:0.05 ≤ P (χ²(17) <  (18 - 1)35.05/σ²M < χ²(0.95)(17))0.05 ≤ P (χ²(17) <  577.57/σ²M < χ²(0.95)(17))0.05 ≤ P (χ²(17) <  577.57/σ²M < 28.412)The above inequality represents the 90%

b. confidence interval estimate for sigma2M.b. 90% con? dence interval estimate for sigma2F:Using the same concept as above, we can write: 0.05 ≤ P (χ²(7) <  (n - 1)s²F/σ²F < χ²(0.95)(7))

Using the table of chi-square values with (n - 1) degrees of freedom, we have:χ²(0.05)(7) =  3.357χ²(0.95)(7) =  14.067

Substituting the values, we have:0.05 ≤ P (χ²(7) <  (8 - 1)18.40/σ²F < χ²(0.95)(7))0.05 ≤ P (χ²(7) <  122.18/σ²F < χ²(0.95)(7))0.05 ≤ P (χ²(7) <  122.18/σ²F < 14.067)

The above inequality represents the 90% confidence interval estimate for sigma2F.

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To calculate a 90% confidence interval estimate for σ2M, the population variance of the amount spent on textbooks by male students, we use the formula below:

[tex]$$\chi_{0.05,17}^2 <\frac{(n - 1)s^2}{\sigma^2} < \chi_{0.95,17}^2$$[/tex]

where n = 18, s2M = 35.05, df = n - 1 = 17, and χα2,

df is the critical value from the chi-squared distribution with df degrees of freedom.

We know that:

[tex]$$\chi_{0.05,17}^2 = 8.909$$[/tex]

and

[tex]$$\chi_{0.95,17}^2 = 31.410$$[/tex]

Substituting these values, we have:

[tex]$$8.909 < \frac{(18-1)(35.05)}{\sigma^2} < 31.410$$[/tex]

Solving for σ2, we have:

[tex]$$\frac{(18-1)(35.05)}{31.410} < \sigma^2 < \frac{(18-1)(35.05)}{8.909}$$[/tex]

Hence, a 90% confidence interval estimate for σ2M is:

[tex]$$(48.704, 194.154)$$b.[/tex]

To calculate a 90% confidence interval estimate for σ2F, the population variance of the amount spent on textbooks by female students, we use the formula below:

[tex]$$\chi_{0.05,7}^2 <\frac{(n - 1)s^2}{\sigma^2} < \chi_{0.95,7}^2$$[/tex]

where n = 8, s2F = 18.40, df = n - 1 = 7, and χα2,

df is the critical value from the chi-squared distribution with df degrees of freedom.

We know that:

[tex]$$\chi_{0.05,7}^2 = 14.067$$[/tex] and

[tex]$$\chi_{0.95,7}^2 = 2.998$$[/tex]

Substituting these values, we have:

[tex]$$14.067 < \frac{(8-1)(18.40)}{\sigma^2} < 2.998$$[/tex]

Solving for σ2, we have:

[tex]$$\frac{(8-1)(18.40)}{2.998} < \sigma^2 < \frac{(8-1)(18.40)}{14.067}$$[/tex]

Hence, a 90% confidence interval estimate for σ2F is:

[tex]$$(7.176, 23.622)$$[/tex]

Therefore, the 90% confidence interval estimate for σ2M,

the population variance of the amount spent on textbooks by male students, is (48.704, 194.154), while the 90% confidence interval estimate for σ2F,

the population variance of the amount spent on textbooks by female students, is (7.176, 23.622).

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The upper and lower readings that would qualify people to participate in the medical study are 113.48 and 126.72 respectively.

Firstly, we'll calculate the 90% confidence interval of the proportion of students who paid for their education by student loans.

For this, we will use the formula:

[tex]$$\hat{p}\pm z\left(\frac{\sqrt{\hat{p}(1-\hat{p})}}{n}\right)$$[/tex]

Where, [tex]$\hat{p}$[/tex] is the sample proportion, n is the sample size, z is the z-score for the level of confidence,

[tex]$1-\alpha$[/tex]

For 90% confidence, the z-score is 1.645 (because the table value of z-score is 1.645 at 90% confidence level).

[tex]\hat{p}=\frac{323}{1404}$$[/tex]

[tex]\hat{p}=0.23$$[/tex]

[tex]\text{Standard Error}= \left(\frac{\sqrt{\hat{p}(1-\hat{p})}}{n}\right)$$[/tex]

[tex]\text{Standard Error}= \left(\frac{\sqrt{0.23(1-0.23)}}{1404}\right)$$[/tex]

[tex]\text{Standard Error}= 0.014$$[/tex]

[tex]\text{Confidence Interval} = \hat{p}\pm z \times \text{Standard Error}$$[/tex]

[tex]\text{Confidence Interval}= 0.23\pm 1.645(0.014)$$[/tex]

[tex]\text{Confidence Interval}= 0.23\pm 0.023$$[/tex]

Confidence Interval = [0.207,0.253]

The 90% confidence interval of the proportion of students who paid for their education by student loans is [0.207,0.253].

Now we will calculate the 95% confidence interval of the mean time.

For this, we will use the formula:

[tex]\bar{X}\pm z\left(\frac{\sigma}{\sqrt{n}}\right)$$[/tex]

Where, [tex]$\bar{X}$[/tex] is sample mean, [tex]$\sigma$[/tex] is population standard deviation, n is sample size, z is the z-score for the level of confidence, [tex]$1-\alpha$[/tex]

For 95% confidence, the z-score is 1.96.

(because the table value of z-score is 1.96 at 95% confidence level).

[tex]\text{Confidence Interval}= \bar{X}\pm z\left(\frac{\sigma}{\sqrt{n}}\right)$$[/tex]

[tex]\text{Confidence Interval}= 7.1\pm 1.96\left(\frac{0.78}{\sqrt{50}}\right)$$[/tex]

[tex]\text{Confidence Interval}= 7.1\pm 0.2199$$[/tex]

Confidence Interval = [6.8801, 7.3199]

The 95% confidence interval of the mean time is [6.8801, 7.3199].

Next, we will find the upper and lower reading that would qualify people to participate in the medical study.

We can do this by calculating the z-scores for the upper and lower percentiles using the standard normal distribution table.

For the lower reading: Since we want to select people in the middle 60% of the population, the lower reading will correspond to the 20th percentile.

Using the standard normal distribution table, we find that the z-score for the 20th percentile is -0.84.

Using the z-score formula, we have:

[tex]z = \frac{x - \mu}{\sigma}$$[/tex]

where, x is the lower reading.

Substituting the given values, we get:-

0.84 = (x - 120) / 8

Solving for x, we get:

[tex]x = (-0.84 \times 8) + 120$$[/tex]

x = 113.48

The lower reading that would qualify people to participate in the medical study is 113.48.

For the upper reading: Since we want to select people in the middle 60% of the population, the upper reading will correspond to the 80th percentile.

Using the standard normal distribution table, we find that the z-score for the 80th percentile is 0.84.

Using the z-score formula, we have:

[tex]z = \frac{x - \mu}{\sigma}$$[/tex]

where, x is the upper reading.

Substituting the given values, we get:

0.84 = (x - 120) / 8

Solving for x, we get:

[tex]x = (0.84 \times 8) + 120$$[/tex]

x = 126.72

The upper reading that would qualify people to participate in the medical study is 126.72.

Therefore, the upper and lower readings that would qualify people to participate in the medical study are 113.48 and 126.72 respectively.

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Answer:

$4.99

Step-by-step explanation:

20 - 5.03 = 14.97

Since that is how much was paid you then divide by how many socks were bought

14.97 ÷ 3 = 4.99

Answer:

4.99/pair

Step-by-step explanation:

Solve

$20 - $5.03 = $14.97 substract

$14.97 ÷ 3 = $4.99 divide

20 - 5.03 = 14.97 = total cost. 14.97÷3 = 4.99/pair.

Therefore, the cost in dollars and cents for each pair of socks is 4.99/pair.

1. This is a valid approach to prove the argument.

2. This is the first step of the contrapositive proof.

3. This statement is true since if one of them is rational, the other one could also be rational or irrational.

4. This statement is true because rational numbers are those numbers that can be expressed as a ratio of two integers.

5. This is true because any rational number can be expressed as a fraction of two integers.

6. This is true since it's the sum of two fractions.

7. This is also true since the sum and product of two integers are always integers.

8. This is true since the product of any non-zero number with another non-zero number is also non-zero.

9. This statement is true since x+y is the quotient of two integers, and since both integers are non-zero, then the quotient is also non-zero and therefore rational.

Therefore, the given argument is correct.

Each step in the argument is logically valid, and the argument follows a correct proof by contrapositive to show that if x is rational and y is rational, then x + y is rational.

The given argument is correct. Let us evaluate each line of the proof and make sure that it is accurate and logical.

Proposition: For every pair of real numbers x and y, if x + y is irrational, then x is irrational or y is irrational

1. We proceed by contrapositive proof.

This is a valid approach to prove the argument.

2. We assume for real numbers x and y that it is not true that x is irrational or y is irrational and we prove that x + y is rational.

This is the first step of the contrapositive proof.

3. If it is not true that x is irrational or y is irrational then neither x nor y is irrational.

This statement is true since if one of them is rational, the other one could also be rational or irrational.

4. Any real number that is not irrational must be rational. Since x and y are both real numbers then x and y are both rational.

This statement is true because rational numbers are those numbers that can be expressed as a ratio of two integers.

5. We can therefore express x as a/b and y as c/d as a, where a, b, c, and d are integers and b and d are both not equal to 0.

This is true because any rational number can be expressed as a fraction of two integers.

6. The sum of x and y is: x + y = a/b + c/d = (ad+bc)/bd

This is true since it's the sum of two fractions.

7. Since a, b, c, and d are integers, the numerator ad + bc and the denominator bd are integers.

This is also true since the sum and product of two integers are always integers.

8. Furthermore since b and d are both non-zero, bd is also non-zero.

This is true since the product of any non-zero number with another non-zero number is also non-zero.

9. Therefore, x + y is a rational number.

This statement is true since x+y is the quotient of two integers, and since both integers are non-zero, then the quotient is also non-zero and therefore rational.

Therefore, the given argument is correct.

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Answer:The probability of the complement of an event is one minus the probability of the event. Since the sum of probabilities of all possible events equals 1, the probability that event A will not occur is equal to 1 minus the probability that event A will occur.

Step-by-step explanation:Complement of an Event: All outcomes that are NOT the event. So the Complement of an event is all the other outcomes (not the ones we want). And together the Event and its Complement make all possible outcomes.

Answer:

7/15

Step-by-step explanation:

7+8=15

there's 7 girls so you have 7/15 chance of getting a girl

Answer:

8/15

Step-by-step explanation:

possibilities/sample size= 8 girls/ 15 students

Answer:

You need to add more information that that. It's too confusing when you put it like that.

Step-by-step explanation:

Answer: girl i'm tryna figure this out too

Step-by-step explanation:

The solution to the triple integral in spherical coordinates is 22000π. This can be obtained by evaluating the integral in three steps: integrating with respect to r, then with respect to θ, and finally with respect to φ.

To change the triple integral to spherical coordinates, we need to express the integrand and the limits of integration in terms of spherical coordinates.

The given integrand is f(x, y, z) = 6x² + y² + z².

In spherical coordinates, the integrand becomes f(r, θ, φ) = 6(rsinθcosφ)² + (rsinθsinφ)² + (rcosθ)².

The limits of integration are as follows:

- The bounds for r are from 0 to 10, as the region Q is bounded by the upper hemisphere x² + y² + z² = 100.

- The bounds for θ are from 0 to π/2, as we are considering the upper hemisphere.

- The bounds for φ are from 0 to 2π, as φ covers a complete revolution around the z-axis.

The triple integral in spherical coordinates is then given by:

∭Q f(r, θ, φ) r² sinθ dr dθ dφ,

which becomes:

∫(φ=0 to 2π) ∫(θ=0 to π/2) ∫(r=0 to 10) [6(rsinθcosφ)² + (rsinθsinφ)² + (rcosθ)²] r sinθ dr dθ dφ.

To solve the given triple integral, we'll start by evaluating the innermost integral with respect to r, then the middle integral with respect to θ, and finally the outer integral with respect to φ.

The integrand is:

[6(rsinθcosφ)² + (rsinθsinφ)² + (rcosθ)²] r² sinθ

First, let's evaluate the innermost integral with respect to r, while treating θ and φ as constants:

∫(r=0 to 10) [6(rsinθcosφ)² + (rsinθsinφ)² + (rcosθ)²] r² sinθ dr

= ∫(r=0 to 10) [6(sin²θcos²φ)r⁴ + (sin²θsinφ)r⁴ + (cos²θ)r⁴] sinθ dr

= ∫(r=0 to 10) [(6sin²θcos²φ + sin²θsin²φ + cos²θ) r⁴] sinθ dr

= [(6sin²θcos²φ + sin²θsinφ + cos²θ) ∫(r=0 to 10) r⁴] sinθ dr

= [(6sin²θcos²φ + sin²θsin²φ + cos²θ) * (10^5/5)] sinθ

= [(6sin²θcos²φ + sin²θsin²φ + cosθ) * 2 × 10⁵] sinθ

Next, let's evaluate the middle integral with respect to θ, while treating φ as a constant:

∫(θ=0 to π/2) [(6sin²θcos²φ + sin²θsin²φ + cos²θ) * 2 × 10⁵] sinθ dθ

= 2 × 10⁵ ∫(θ=0 to π/2) [6sin²θcos²φ + sin²θsin²φ + cos²θ] sinθ dθ

= 2 × 10⁵ [2/3cos²φ + 1/4 + 1/3]

= 2 × 10⁵ [2/3cos²φ + 7/12]

Finally, let's evaluate the outer integral with respect to φ:

[tex][\int_{0}^{2\pi} 2\times10^5 \left( \frac{2}{3}\cos^2\phi + \frac{7}{12} \right) d\phi \\\\= 2\times10^5 \left( \frac{2}{3}\pi + \frac{7}{12}(2\pi) \right)][/tex]

= 22π × 10000

= 22000π

Therefore, the solution to the given triple integral is 22000π.

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Complete question :

Change the triple integral to spherical coordinates: SIS 6x2 + y2 + z2) av (༴ AV Where Q is bounded by the upper hemisphere: x2 + y2 +22=100 : 21 10 ("S", p's p. sino dpdooo 2T pº sino dododo 21 10 2 3 sino doopde 0 0 0 10 ["S" p2 sino apdoce

Answer:

there is only one zero

Step-by-step explanation: and if it easy why you didint do it just ascking

Answer:

1. B. $20

2.C. 19%

3. D. not here

Step-by-step explanation:

1.  info we know

25+45+10+25+15+10 = 130

we need 20 more dollars

2.  25/130   x/100

cross multiply

2500/ 130x

divide

19 = x

3.  same steps as last time

35/130 x/100

3500/130x

29%

Answer:

523 inches

Step-by-step explanation:

got it right on edg

There are four degrees of freedom for the chi-square goodness-of-fit test in this study. So, correct option is B.

For a chi-square goodness-of-fit test in the context of testing the fairness of a spinner with five sections, the number of degrees of freedom can be determined by subtracting 1 from the number of categories being tested.

In this case, since the spinner has five sections, there are five categories. Therefore, the degrees of freedom for the chi-square goodness-of-fit test would be:

Degrees of freedom = Number of categories - 1

= 5 - 1

= 4

Degrees of freedom represent the number of values in the final calculation of the chi-square test statistic that are free to vary. It determines the critical values and the distribution of the test statistic.

In the case of the chi-square goodness-of-fit test, the test compares the observed frequencies in each category with the expected frequencies under the assumption of fairness. By comparing these frequencies, the test determines if there is a significant deviation from the expected distribution, indicating unfairness in the spinner.

So, correct option is B.

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Answer:

9

Step-by-step explanation:

I'm not sure, the answer is still 9, 1+2 is 3 so

6/2(3)

and 6/2 simplified is 3

so 3(3) is 9

Answer:

9

Step-by-step explanation:

1. Simplify the parantheses

(1+2) = 3

2. Turn 3 into a fraction

3 = 3/1

3. Multiply the fractions

6   x    3    =  18

2    x   1         2

4. What is 18/2?

18/2 = 9

Hiii, so I REALLY want to rank up, and I just need 5 more Branliests, so if you liked my answer, can you please give me one? Thank you so much, and thanks for the points!!

Answer:

hola

Como te amo hermoso

Step-by-step explanation:

Te conozco a alguien para mi amor todo

Answer:

D

Step-by-step explanation:

A and C are what we've always been taught so it's not them. B is just side time side so with a square that works. D is the only one what I'm not sure of or doesn't seem familiar. So personally I'd say D. Good luck!

Answer:

D

explanation

a finds area of rectangle

b area of square

c of rhombus

They have different end behavior as x approaches -∞ but the same end behavior as x approaches ∞. Option A is correct.

Given that,
Function g is a decreasing exponential function with a y-intercept of 5 and no x-intercept.

Functions are the relationship between sets of values. e g y=f(x), for every value of x there is its exists in a set of y. x is the independent variable while Y is the dependent variable.

The function which is in format f(x) = [tex]e^X[/tex]   where, a is constant and x is variable,  the domain of this exponential function lies   (-∞, ∞).  

Here, the solution shown in the graph implies that they have different end behavior as x approaches -∞ but the same end behavior as x approaches ∞.

Thus, they have different end behavior as x approaches -∞ but the same end behavior as x approaches ∞. Option A is correct.

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Answer:

y=3x+25

Step-by-step explanation:

For the equation y=3x+5, the slope is 3. Since the line will be parallel to it then this is the slope of the new line as well. Use the point-slope form to write the equation, then simplify and convert into the slope intercept form.

(y-13)=3(x--4)

y-13=3x+12

y=3x+25

Answer:

4,370

Step-by-step explanation:

.....just round the tenths place (6) up sense 8 is above 5 it rounds up so

4,370

sorry for bad explanation

a) The standard error of the mean can be calculated using the formula:

standard error = sample standard deviation / √(sample size).

Given a sample size (n) of 51 and a sample standard deviation (s) of 1.4, we can compute the standard error as follows:

Standard error = 1.4 / √51 ≈ 0.1967 (rounded to four decimal places).

b) To find the critical value of t^* for a 99% confidence interval, we need to consider the degrees of freedom. Since we have a sample size of 51, the degrees of freedom is n-1 = 51-1 = 50. Using a t-distribution table or calculator, the critical value for a 99% confidence interval with 50 degrees of freedom is approximately ±2.680.

c) To construct a 99% confidence interval for µ, we can use the formula:

Confidence interval = sample mean ± (critical value * standard error).

Using the given sample mean of 87 and the standard error calculated in part a, the confidence interval can be calculated as follows:

Confidence interval = 87 ± (2.680 * 0.1967) ≈ 87 ± 0.5278

d) Since the confidence interval obtained in part c does not include the hypothesized value of 82, we can reject the null hypothesis (H_o: μ = 82) at α = 0.010. The hypothesized value of 82 falls outside the confidence interval, providing evidence to suggest that the true population mean is different from 82.

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The probability of a positive test, given breast cancer, is 223/303, while the probability of a negative test, given no breast cancer, is 684/697.

To find the probability of a positive test, given that the patient has breast cancer, we divide the number of true positive cases (223) by the total number of patients with breast cancer (303). This gives us a probability of 223/303.

The formal term used to describe this probability measure is conditional probability or the probability of an event A occurring given that event B has already occurred. In this case, the positive test is event A, and having breast cancer is event B.

Similarly, to find the probability of a negative test, given that the patient is breast cancer-free, we divide the number of true negative cases (684) by the total number of patients without breast cancer (697). This gives us a probability of 684/697.

The formal term used to describe this probability measure is also conditional probability, where the negative test is event A, and not having breast cancer is event B.

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Answer:

30 x 4 + 4 x 4 and 4 x 4 + 4 x 30

Step-by-step explanation:

Answer:

Option 1: 4h

Step-by-step explanation:

perimeter of square = 4 × length

h × 4 = 4h

Answer:

B

Step-by-step explanation:

it's b

m= d-10

The equation which best represents the total amount in the account, m, when starting with d dollars is

An equation is the statement of two expressions located on two sides connected with an equal to sign. The two sides of an equation is usually called as left hand side and right hand side.

Given that,

A bank charges a $10 fee to open an account.

Let the account is started with d dollars.

From the amount d, $10 will be taken as an opening fee.

Remaining balance = d - 10

So if m represents the total amount in the account, the,

m = d - 10

Hence the total amount in the account, m, can be represented as m = d - 10, if the account is started with d dollars.

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Answer:

28yd^2

Step-by-step explanation:

the answer is 28 yards squared bud :)

Answer:

385 are math quetions.

Step-by-step explanation:

70% of 550 is 385.

The final simplified expression is [tex](sin(t))^2[/tex]is the correct answer.

The given expression is [tex]sin^2(t) / sin^2(t) + cos^2(t).[/tex]

Simplify [tex]sin^2(t)/sin^2 (t) + cos^2(t)[/tex] to an expression involving a single trig function with no fractions:

By using the identity[tex]sin^2 (t) + cos^2 (t) = 1[/tex] we can write, [tex]sin^2(t)/sin^2 (t) + cos^2(t) = sin^2(t)/(sin^2(t) + cos^2(t))[/tex]

Now using the identity [tex]csc^2 (t) = 1/sin^2 (t)[/tex] we get, [tex]sin^2(t)/(sin^2(t) + cos^2(t))= 1/csc^2(t) = (sin(t))^2[/tex]

The final simplified expression is [tex](sin(t))^2.[/tex]

Trigonometry is the study of relationships between angles and sides of triangles. It finds applications in a variety of fields like engineering, physics, architecture, etc. Trigonometric ratios, identities, and functions are the main concepts of trigonometry. Trigonometric ratios of an angle are ratios of the lengths of two sides of a triangle containing that angle. They are sine, cosine, tangent, cosecant, secant, and cotangent, which are abbreviated as sin, cos, tan, csc, sec, and cot, respectively. The primary trigonometric identity is [tex]sin^2 (t) + cos^2 (t) = 1.[/tex]

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