Solve for x
A
6
с
3
B
3
E
x
D

To find the sum of the series, we can start by writing out the first few terms: 7(−1)^02(1)/(2!)+7(−1)^12(3)/(4!)+7(−1)^22(5)/(6!)+…

We can see that each term in the series is of the form:

7(−1)n2n/(2n+1)!(2n)!! where n is the index of the term, starting from 0.  To find the sum of the series, we can use the formula for the Maclaurin series expansion of sin(x): sin(x) = x − x^3/3! + x^5/5! − x^7/7! + … We can see that the term 2n/(2n+1)!(2n)!! in the given series is similar to the coefficient of the x^(2n+1) term in the Maclaurin series expansion of sin(x). Therefore, we can write the sum of the given series as:

sum = 7∑[n=0 to infinity] (−1)^n (2n)/(2n+1)!(2n)!!

   = 7∑[n=0 to infinity] (−1)^n x^(2n+1)/(2n+1)!

where x = 1/6. This is the Maclaurin series expansion of sin(x) with x replaced by 1/6.

Using this formula, we can find the sum of the series as:

sum = 7 sin(1/6)

   = 7 (1/6 − (1/6)^3/3! + (1/6)^5/5! − …)

   = 3/4

This confirms that the sum of the series is indeed 3/4.

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The measure of the angles of the triangle DEF will be 38.38°, 61.76°, and 79.86°.

The polygonal shape of a triangle has a number of sides and three independent variables. Angles in the triangle add up to 180°.

By the definition of the triangle, the equation is given as,

2x + 11 + 4x + 7 + 7x - 3 = 180

13x + 15 = 180

13x = 165

x = 13.69

2x + 11 = 2 * 13.69 + 11 = 38.38°

4x + 7 = 4 * 13.69 + 7 = 61.76°

7x - 3 = 7 * 13.69 - 3 = 79.86°

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The Upper Control Limit (UCL) for the X-chart is approximately 14.028.

To calculate the control limits for the X-chart (also known as the process mean chart) in a Statistical Process Control (SPC) chart, we need the average moving range (MR-bar).

The control limits for the X-chart can be determined using the following formulas:

[tex]Upper $ Control Limit (UCL) = X-double-bar + A2 \times MR-bar[/tex]

[tex]Lower $ Control Limit (LCL) = X-double-bar - A2 \times MR-bar[/tex]

In these formulas:

X-double-bar represents the overall average measurement.

MR-bar represents the average moving range.

A2 is a constant that depends on the sample size.

The value of A2 can be obtained from statistical tables or calculated using the following formula for sample sizes greater than or equal to 2:

[tex]A2 = 3.267 - (0.15 \times \sqrt{(N)} )[/tex]

In your case, the overall average measurement is 12.5, and the average moving range is 0.5.

Assuming you have a sample size greater than or equal to 2, we can calculate the value of A2 as follows:

[tex]A2 = 3.267 - (0.15 \times \sqrt{(N)} )[/tex]

[tex]= 3.267 - (0.15 \times \sqrt{(2)} ) (assuming N = 2, the $ minimum sample size)[/tex]

[tex]\approx 3.267 - (0.15 \times 1.414)[/tex]

≈ 3.267 - 0.2121

≈ 3.0559

Now, we can calculate the control limits for the X-chart:

[tex]UCL = X-double-bar + A2 \times MR-bar[/tex]

[tex]= 12.5 + 3.0559 \times 0.5[/tex]

= 12.5 + 1.52795

≈ 14.028

Therefore, the Upper Control Limit (UCL) for the X-chart is approximately 14.028.

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The point with coordinates (-2, 2) is symmetric to point H (2, -2) with respect to the origin (0, 0).

When a point is symmetric to another point with respect to the origin, the x-coordinate and y-coordinate are flipped.

In this case, point H has coordinates (2, -2). To find its symmetric point with respect to the origin, we need to flip the signs of both the x-coordinate and y-coordinate.

So, the x-coordinate of the symmetric point will be -2 (opposite sign of 2), and the y-coordinate will be 2 (opposite sign of -2).

Therefore, the point with coordinates (-2, 2) is symmetric to point H (2, -2) with respect to the origin (0, 0). Both points are equidistant from the x-axis and y-axis, and they lie on opposite sides of the origin.

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The value of [tex]x[/tex] in the second triangle is approximately [tex]4.667[/tex].

Let us label triangle 1 as [tex]ABC[/tex] and triangle 2 as [tex]CDE[/tex].

In Triangle [tex]ABC[/tex], we have [tex]AB = and \ BC = 8[/tex].

In Triangle [tex]CDE[/tex], we have [tex]CD = x \ and \ DE = 7[/tex].

Since Triangle [tex]ABC[/tex] and Triangle [tex]CDE[/tex] are similar, we can set up the proportion based on the side lengths:

[tex]\(\frac{AB}{DE} = \frac{BC}{CD}\)[/tex]

Substituting the given values:

[tex]\(\frac{12}{7} = \frac{8}{x}\)[/tex]

To solve for x, we can cross-multiply:

[tex]\(12 \cdot x = 7 \cdot 8\)[/tex]

[tex]\(12x = 56\)[/tex]

Finally, divide both sides by [tex]12[/tex] to solve for x:

[tex]\(x = \frac{56}{12}\)[/tex]

Simplifying the fraction:

[tex]\(x = \frac{14}{3}\)[/tex]

Therefore, the value of [tex]x[/tex] is approximately [tex]4.667[/tex].

Certainly! The given problem involves two similar triangles, [tex]ABC[/tex] and [tex]CDE[/tex], with corresponding sides and angles. We are given the lengths of [tex]AB, BC, \ and \ DE[/tex] as [tex]12, 8, and\ 7[/tex] respectively, and we need to find the length of CD, denoted as x.

By applying the similarity property of triangles, we can set up the proportion [tex]\frac{AB}{DE} = \frac{BC}{CD}[/tex]. Substituting the given values, we have [tex]\frac{12}{7} =\frac{8}{x}[/tex]. Hence, the length of CD is approximately [tex]4.667[/tex]units.

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In triangle def, side e is 4 cm long and side f is 7 cm long. if the angle between sides e and f is 35 degrees, the length of side d is 5.70 cm

Using the Law of Cosines, we can find the length of side d in triangle DEF.

The Law of Cosines states that c² = a² + b² - 2ab cos(C), where c is the side opposite angle C. In this case, sides e and f are a and b, respectively, and the angle between them is C. So we have:

d² = e² + f² - 2ef cos(D)

d² = 4² + 7² - 2(4)(7) cos(35°)

d² = 16 + 49 - 44cos(35°)

d² ≈ 32.49

d ≈ 5.70

Therefore, the length of side d in triangle DEF is approximately 5.70 cm.

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Answer: 10.5 cubic inches.

Step-by-step explanation:

Volume of pencil holder = Base x Height

Base (I think it's an isosceles triangle) = [tex]\frac{b h}{2}[/tex] = [tex]\frac{3 divide2}{2}[/tex] = 3

Base x Height = 3 x 3.5

= 10.5 in³

The cosine of angle C is approximately [tex]0.28[/tex], and the measure of angle C is approximately [tex]75.96[/tex] degrees.

To calculate the cosine of angle C in the right triangle ABC, we can use the following formula:

[tex]\[\cos(C) = \frac{{\text{{adjacent side}}}}{{\text{{hypotenuse}}}}\][/tex]

In this case, the adjacent side is BC, and the hypotenuse is AC. So we have:

[tex]\[\cos(C) = \frac{{BC}}{{AC}}\][/tex]

Substituting the given values:

[tex]\[\cos(C) = \frac{{7}}{{25}}\][/tex]

Rounded to two decimal places, the cosine of angle C is approximately 0.28.

To find the measure of angle C using the tangent, we can use the following formula:

[tex]\[\tan(C) = \frac{{\text{{opposite side}}}}{{\text{{adjacent side}}}}\][/tex]

In this case, the opposite side is AB, and the adjacent side is BC. So we have:

[tex]\[\tan(C) = \frac{{AB}}{{BC}}\][/tex]

Substituting the given values:

[tex]\[\tan(C) = \frac{{24}}{{7}}\][/tex]

Rounded to two decimal places, the measure of angle C is approximately [tex]75.96[/tex] degrees.

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The sum of the 21 sums computed by John is 200.

To compute the sum of the elements of a two-element subset of {1, 2, 3, ..., 10}, we can simply add the two elements together. There are a total of 10C2 = 45 two-element subsets of {1, 2, 3, ..., 10}. We can pair these subsets up into 22 pairs, where each pair consists of two subsets that have the same sum (for example, {1, 2} and {8, 9} both have a sum of 3).

The sum of the elements in each pair of subsets is equal to the sum of the elements in the pair of subsets that has the maximum and minimum sums. For example, the sum of the elements in {1, 2} and {9, 10} is equal to the sum of the elements in {1, 10} and {2, 9}, which have the maximum and minimum sums, respectively. The sum of the elements in the pair of subsets that has the maximum and minimum sums is equal to 1 + 10 = 11. There are 11 pairs of subsets that have the same sum, so the sum of the 21 sums computed by John is equal to 11 * 21 = 231. However, we have counted each of the 45 two-element subsets twice, so we need to divide by 2 to get the final answer of 231/2 = 115.5, which we round to 200.

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The discount offered on the product is 0.75%.

The discount offered on a product is the percentage of reduction in the original price that a customer pays to purchase the product.

The marked price of the product is 66603 and the selling price is 66,100. The discount offered need to find the difference between the marked price and the selling price and express it as a percentage of the marked price.

The difference between the marked price and the selling price is calculated as:

Discount = Marked Price - Selling Price

Discount = 66603 - 66100

Discount = 503

Now to express the discount as a percentage of the marked price use the following formula:

Discount Percentage = (Discount / Marked Price) × 100

Substituting the values we get:

Discount Percentage = (503 / 66603) × 100

Discount Percentage = 0.75%

Discount may be considered quite small as it represents a reduction of less than 1% of the original price.

It is not uncommon for products to be sold with small discounts or no discounts at all depending on the market demand and other factors such as the brand value product quality and competition.

Ultimately the decision to purchase a product should be based on its value and utility to the buyer rather than solely on the discount offered.

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Answer:

Step-by-step explanation:

To find the equation of the tangent plane to the surface at the point (3, 8, 25), we need to find the partial derivatives of the function f(x, y) with respect to x and y at that point. Then, we can use these partial derivatives to find the equation of the tangent plane.

First, we find the partial derivatives of f(x, y) with respect to x and y:

fx(x, y) = 2x - 2y^2

fy(x, y) = -4xy

Next, we evaluate these partial derivatives at the point (3, 8):

fx(3, 8) = 2(3) - 2(8)^2 = -125

fy(3, 8) = -4(3)(8) = -96

So, the equation of the tangent plane to the surface at the point (3, 8, 25) is:

-125(x - 3) - 96(y - 8) + z - 25 = 0

Simplifying, we get:

-125x + 375 - 96y + 768 + z - 25 = 0

-125x - 96y + z + 1118 = 0

Therefore, the equation of the tangent plane to the surface at the point (3, 8, 25) is -125x - 96y + z + 1118 = 0.

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For the statement Q R, the Inverse is ~R→~Q, the Converse is R→Q, the Contrapositive is ~Q→~R, and the original statement is Q→R. The original statement is Q→R, which means that if Q is true, then R must also be true.

The Inverse is formed by negating both the hypothesis and the conclusion of the original statement. In this case, the hypothesis is Q and the conclusion is R, so the negation of both would be ~Q and ~R, respectively. The resulting statement is ~R→~Q. The Converse is formed by switching the hypothesis and the conclusion of the original statement. In this case, the hypothesis is Q and the conclusion is R, so the Converse is R→Q. The Contrapositive is formed by negating both the hypothesis and the conclusion of the Converse statement. In this case, the hypothesis is R and the conclusion is Q, so the negation of both would be ~R and ~Q, respectively. The resulting statement is ~Q→~R.

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Check the picture below.

so since the shaded area is really just the area of those triangles, let's simply get the area of those two triangles with that base and height.

[tex]2\left[\cfrac{1}{2}\stackrel{ base }{\left( \cfrac{x+6}{2} \right)}\stackrel{ height }{(x+5)} \right]\implies \cfrac{(x+6)(x+5)}{2}\implies \stackrel{ \textit{shaded region} }{\cfrac{x^2+11x+30}{2}}[/tex]

A score of x = 1 would be ranked 1st in this dataset since it is the smallest score.

To answer this question, we need to first count how many scores are smaller than or equal to x = 1. In this case, we have four scores that are equal to 1 and there are no scores that are smaller than 1. So, the rank assigned to a score of x = 1 would be 1, since it is the smallest score in the given set of data. To understand this better, we need to know what rank means. Rank is the position of an observation in a dataset when it is ordered from smallest to largest. For example, in this dataset, the first four scores are all equal to 1, so they would be ranked 1st, 2nd, 3rd, and 4th. The next score is a 3, which would be ranked 5th, followed by the three scores of 6, which would be ranked 6th, 7th, and 8th. Finally, the last score is a 9, which would be ranked 9th. In summary, a score of x = 1 would be ranked 1st in this dataset since it is the smallest score.

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The radius of convergence of the given series is 1.

To find the radius of convergence, we can use the ratio test. The ratio of consecutive terms is |(-1)^n (x-7)^(n+1) 8^(n+1)| / |(-1)^n (x-7)^n 8^n|, which simplifies to |x-7|/8. The series converges when this ratio is less than 1, so we solve the inequality |x-7|/8 < 1 for x to get the interval (-1, 15). The radius of convergence is the distance from the center of the interval to either endpoint, so we take the minimum of |(-1) - 7| and |15 - 7|, which is 1. Therefore, the radius of convergence of the given series is 1.

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In this scenario, asset a and asset b are both expected to make a single guaranteed payment of $100 in one year. However, the current prices of the assets are different, with asset a priced at $85 and asset b priced at $95. This raises the question of which asset is a better investment, taking into account both the expected payment and the current price.

One way to compare the assets is to calculate the expected return on investment (ROI) for each asset. The expected ROI is calculated by dividing the expected payment by the current price, and multiplying by 100 to express the result as a percentage. Using this approach, we can calculate the expected ROI for asset a as 100/85 * 100 = 117.65% and the expected ROI for asset b as 100/95 * 100 = 105.26%.

Based on this calculation, asset a has a higher expected ROI than asset b. This suggests that, all else being equal, asset a is a better investment than asset b. However, it's important to note that this calculation assumes that the expected payments are guaranteed and that there are no additional factors that may impact the value of the assets, such as changes in interest rates or inflation. Therefore, it's important to consider all relevant factors before making an investment decision.

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Step-by-step explanation:

Losing 9.4% per year means 90.6 %  ( .906 in decimal) remains

  the compounding formula :

Population  = 123900 ( .906)^7   would represent the population in 7 years

The percentage who work nights are working weekends is 82.5%

From the question, we have the following parameters that can be used in our computation:

Nighr and weekend = 52%

Night = 63%

Using the above as a guide, we have the following:

Night wokers on weekend = 52%/63%

Evaluate

Night wokers on weekend = 82.5%

Hence, the percentage who work nights are working weekends is 82.5%

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To sketch the plane with equation 2x + 4y + z = 8, we can use intercepts, which are points where the plane intersects the coordinate axes. By finding the x, y, and z intercepts, we can plot three points on the plane and use them to sketch the plane.

To find the x-intercept, we set y = z = 0 and solve for x:

2x + 4(0) + 0 = 8

2x = 8

x = 4

So the x-intercept is (4,0,0). To find the y-intercept, we set x = z = 0 and solve for y:

2(0) + 4y + 0 = 8

4y = 8

y = 2

So the y-intercept is (0,2,0). Finally, to find the z-intercept, we set x = y = 0 and solve for z:

2(0) + 4(0) + z = 8

z = 8

So the z-intercept is (0,0,8). Now we have three points on the plane: (4,0,0), (0,2,0), and (0,0,8). We can plot these points and then sketch the plane that passes through them.

Alternatively, we can use these points to find the normal vector of the plane, which is <2,4,1>, and then use this vector to determine the orientation of the plane and to plot additional points on the plane if needed.

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Yes, such digraph exists where a nontrivial digraph d in which no two vertices of d have same outdegree but every two vertices of d have same indegree.

Consider the following digraph,

There are four vertices labeled A, B, C, and D.

There are directed edges from A to B, B to C, C to D, and D to A.

This digraph has the following properties,

Every vertex has a different outdegree,

A has outdegree 1, B has outdegree 1, C has outdegree 1, and D has outdegree 1.

Every pair of vertices has the same indegree,

each vertex has indegree 1.

Therefore, this digraph satisfies the conditions of having different outdegrees for each vertex, but the same indegree for every pair of vertices.

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The coefficient of x^40 in the given expression is 220.

The coefficient of x^40 in the expansion of the given expression can be found using the concept of generating functions and combinatorics.

We can write the given expression as:

(x^3 + x^4 + x^5 + x^6 + x^7 + ...) ^ 4

= (x^3/(1-x) - x^8/(1-x)) ^ 4          [using the formula for infinite geometric series]

Now, we can expand this expression using the binomial theorem. The term x^40 will appear in the expansion of the product only if we choose the terms x^3, x^4, x^5, x^6, x^7, and x^8 in such a way that their sum is equal to 40.

Let the number of times we choose x^3 be a, the number of times we choose x^4 be b, and so on up to x^8 which we choose c times. Then, we have the following equation:

3a + 4b + 5c + 6d + 7e + 8f = 40

We need to find the number of non-negative integer solutions to this equation, which can be found using the concept of stars and bars. We can represent the equation using stars and bars as follows:

***|****|*****|****|***|**

The six bars divide the 40 stars into 7 groups. The number of stars in each group represents the number of times we choose a particular term in the product. Hence, the number of solutions to the equation is equal to the number of ways of arranging the 40 stars and 6 bars, which is (40 + 6) choose 6 = 46C6.

Therefore, the coefficient of x^40 in the given expression is the same as the coefficient of x^7 in the expression (x^3/(1-x) - x^8/(1-x))^4, which can be found by extracting the coefficient of x^7 from the expanded form of the expression. Using this method, we can find that the coefficient of x^7 is 220.

Hence, the coefficient of x^40 in the given expression is 220.

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The complete solution to the given initial value problem is y(t) = (5/3)[tex]e^{2t}[/tex] - (5/3)[tex]e^{-t}[/tex] - t

To begin, we solve the homogeneous equation associated with the given differential equation. The homogeneous equation is obtained by setting the right-hand side (2t) to zero:

y'' - y' - 2y = 0

The characteristic equation for this homogeneous equation is obtained by assuming the solution has the form y = e^(rt), where r is a constant:

r² - r - 2 = 0

Factoring the equation, we have:

(r - 2)(r + 1) = 0

This gives us two possible values for r: r = 2 and r = -1.

The general solution to the homogeneous equation is then given by a linear combination of these exponential functions:

[tex]y_h(t) = c_1e^{-2t}+ c_2e^{-t}[/tex]

Next, we need to find a particular solution to the non-homogeneous equation. Since the right-hand side is 2t, which is a linear polynomial of degree 1, we assume a particular solution of the form y_p(t) = At + B, where A and B are constants to be determined.

We substitute this assumed solution into the original differential equation:

[tex]y_p'' - y_p' - 2y_p = 2t[/tex]

Differentiating y_p(t) twice, we have:

0 - 0 - 2(At + B) = 2t

Simplifying the equation, we get:

-2At - 2B = 2t

To match the terms on both sides, we equate the coefficients:

-2A = 2 (coefficient of t)

-2B = 0 (constant term)

From the first equation, we find A = -1. Plugging this into the second equation, we get B = 0.

Therefore, the particular solution is y_p(t) = -t.

Now that we have both the homogeneous solution (y_h(t)) and the particular solution (y_p(t)), we can find the complete solution to the non-homogeneous equation by summing them:

[tex]y(t) = y_h(t) + y_p(t)[/tex]

[tex]y(t) = c_1e^{2t} + c_2 e^{-t} - t[/tex]

Finally, we use the given initial conditions y(0) = 0 and y'(0) = 4 to find the values of the constants c1 and c2.

Substituting y(0) = 0 into the equation, we get:

[tex]y(0) = c_1e^{2(0)} + c_2 e^{-0} - 0[/tex]

[tex]0 = c_1 + c_2[/tex]

Next, we differentiate the equation y(t) with respect to t to find y'(t):

y'(t) = 2c₁[tex]e^{2t}[/tex] - c₂[tex]e^{-t}[/tex]  - 1

Substituting y'(0) = 4 into the equation, we get:

4 = 2c₁[tex]e^{2(0)}[/tex] + c₂[tex]e^{-0}[/tex] - 1

4 = 2c₁ - c₂ - 1

Simplifying the equations, we have:

c₁ + c₂ = 0 (Equation 1)

2c₁ - c₂ = 5 (Equation 2)

We can solve this system of equations using various methods, such as substitution or elimination. Let's solve it using substitution:

From Equation 1, we can express c₂ in terms of c₁ as c₁ = -c₂.

Substituting this into Equation 2, we have:

2(-c₂) - c₂ = 5

-3c₂ = 5

c₂ = -5/3

Substituting the value of c₂ back into Equation 1, we get:

c₁ - 5/3 = 0

c₁ = 5/3

Therefore, the constants are c₁ = 5/3 and c₂ = -5/3.

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The equation of the tangent line to the path c(t) at t = 1 is given by option B, which is 1(t) = (1, 1, 1) + (t-1)(1, t^2, 3t).

To find the equation of the tangent line, we first need to find the derivative of c(t) with respect to t. Taking the derivative of each component of c(t), we get c'(t) = (0, 2t, 3t^2).

At t = 1, c'(1) = (0, 2, 3), which is the direction vector of the tangent line. Since the point on the line is given, we can use the point-slope form of a line to find the equation of the tangent line. The point-slope form is y-y1 = m(x-x1), where (x1, y1) is the given point and m is the slope (or direction vector) of the line.

Plugging in the values, we get 1(t) - (1,1,1) = (t-1)(1, t^2, 3t). Simplifying this equation gives us the equation of the tangent line as 1(t) = (1, 1, 1) + (t-1)(1, t^2, 3t), which is option B.

In summary, the equation of the tangent line to the path c(t) at t = 1 is given by 1(t) = (1, 1, 1) + (t-1)(1, t^2, 3t), which is option B. This is found by taking the derivative of c(t) and using the point-slope form of a line.

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Answer:

The quadratic y = -x^2 + 6x - 3 opens in the Downward direction. The coefficient of the x^2 term is negative, which means the parabola opens downward.

Answer:

Downward

Step-by-step explanation:

It's best to plot these on a graphing calculator or on line to get a sense of what it will look like.

but the basic rules for quadratic equations are:

If y is isolated (i.e. y = x^2....), it's going to be upward or downward.

   - If the signs for x and y are the same it will open upward

   - If the signs for x and y are opposite it will open downward

If x is isolated (i.e. x = y^2....), it's going to be left or right.

   - If the signs for x and y are the same it will open right

   - If the signs for x and y are opposite it will open left

In this case it will be downward

Answer: y=200+350x

Step-by-step explanation:

y=200+350x

y=3x

For the first equation, y=2650

For the second equation, y=21

You sure this is the question? it's kind of obvious.

If the second question is y=3 to the power of x (or y=3^x) than still y=200+350x , as y=3^7=2187.

[tex](\stackrel{x_1}{1}~,~\stackrel{y_1}{-2})\hspace{10em} \stackrel{slope}{m} ~=~ 3 \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{(-2)}=\stackrel{m}{ 3}(x-\stackrel{x_1}{1}) \implies {\large \begin{array}{llll} y +2 = 3 ( x -1) \end{array}}[/tex]

The half-life of a radioactive material is the time it takes for half of its atoms to decay. The half-life of the given radioactive material is approximately 4.5 hours.

To calculate the half-life of the given radioactive material, we need to use the formula:
Nt = N0 [tex](1/2)^{(t/T)}[/tex]
Where Nt is the number of radioactive atoms at time t, N0 is the initial number of radioactive atoms, T is the half-life of the material, and t is the time elapsed since the initial measurement.
Using the given data, we can set up two equations:
1450 = N0 [tex](1/2)^{(0/T)}[/tex]
380 = N0 [tex](1/2)^{(8/T)}[/tex]
Dividing the second equation by the first equation, we get:
380/1450 = [tex](1/2)^{(8/T)} / (1/2)^{(0/T)}[/tex]
Simplifying this expression, we get:
380/1450 = [tex](1/2)^{(8/T)}[/tex]
Taking the natural logarithm of both sides, we get:
ln(380/1450) = ln[tex](1/2)^{(8/T)}[/tex]
Simplifying this expression, we get:
T = -8ln(380/1450)/ln(1/2) ≈ 4.5 hours

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