Find the 15th term of the geometric sequence 8,32,128

Answer:

3071.6 ft

Step-by-step explanation:

In this situation,

We can use sine law,

in which Sin angle = opposite side/ hypotenuse

Similarly

In triangle ABC

SIn A=BC/AC

Sin 19°=1000/AC

Doing criss cross multiplication:

AC= 1000/(Sin 19°)

AC=3071.55 ft

Therefore, the distance I ski down the mountain is 3071.6 ft

The diagram is below.

==============================================

Work Shown:

sin(angle) = opposite/hypotenuse

sin(19) = 1000/x

xsin(19) = 1000

x = 1000/sin(19)

x = 3071.553487

x = 3071.6

Refer to the diagram below.

Note that the angle of depression ACD is equal to the angle of elevation BAC because of the alternate interior angles theorem.

Answer:

(x + 5)² + (y – 1)² = 25

Step-by-step explanation:

Note that the general equation of a circle is (x – h)² + (y – k)² = r², where (h, k) represents the location of the circle's center, and r represents the length of its radius.

The circle is 10 units in diamater, so the radius is half this: r=10/2=5.

The circle goes from -10 to 0 along the x-axis, so the x-coordinate of its centre is halfway between this: h=(-10+0)/2=-10/2=-5

The circle goes from -4 to 6 along the y-axis, so the y-coordinate of its centre is halfway between this: k=(-4+6)/2=-2/2=1

Now that we know r, h, and k, we can sub these into the general formula to get our equation:

(x - (-5))² + (y – 1)² = 5²

(x + 5)² + (y – 1)² = 25

Note that the longest segment in the shapes shown is DC (Option B).

The longest side of a triangle is opposite to greatest angle.

To determine the longest side in a triangle, compare the lengths of all three sides. The side with the greatest length is the longest side.

You can use a ruler or a measuring tool to measure the lengths of the sides or compare the numerical values if they are provided.

In this case,

∠A = ∠DBA = 60°

So ∠ ABD is an equilateral triangle.

So, AB = BD = AD

Since

AB = BC

Then

∠BDC = ∠C ∠ 38°

so ∠DBC > 90°

This means that DC is the longest side.

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After 8 visits: General Admission: Each person would pay $200. Annual Public Session Pass: Each person would pay $299.

After 10 visits: General Admission: Each person would pay $250. Annual Public Session Pass: Each person would still pay $299.

To determine when it is better to have a session pass versus just paying general admission, we need to compare the costs for each option.

General Admission with skate rental: $25

Annual Public Session Pass with skate rental: $299

After 8 visits:

For General Admission, the cost per visit would be $25 per visit.

Total cost for 8 visits: $25 x 8 = $200

For the Annual Public Session Pass, the cost is a one-time payment of $299, regardless of the number of visits. Therefore, after 8 visits, the cost remains the same at $299.

Therefore, after 8 visits, it would be more cost-effective to have the Annual Public Session Pass as the cost per person would be $299.

After 10 visits:

For General Admission, the cost per visit remains at $25 per visit.

Total cost for 10 visits: $25 x 10 = $250

For the Annual Public Session Pass, the cost remains the same at $299, regardless of the number of visits.

Therefore, after 10 visits, it would still be more cost-effective to have the Annual Public Session Pass as the cost per person would still be $299.

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The following question may be like this:

When is it better to have a session pass versus just paying general admission? After 8 visits: How much would each person pay? Show your work for calculations. Session Pass______? General Admission______?After 10 visits: How much would each person pay? Show your work for calculations.Session pass _____? General Admission_____?

GENERAL ADMISSION WITH SKATE RENTAL  $25

ANNUAL PUBLIC SESSION PASS WITH SKATE RENTAL $299

LOCKER RENTAL-ONE TIME USE  $5

HELMET RENTAL  $5

SKATE HELPER-PER HOUR $15

Answer:

az

Step-by-step explanation:

The equivalent expressions for this problem are given as follows:

Equivalent expressions are the expressions that have the same result, hence we must simplify each expression.

The first expression is given as follows:

(4x³ + 7x - 4) - (2x³ - x - 8).

Simplifying the like terms, we have that:

Hence it is equivalent to expression B.

The second expression is simplified as follows:

[tex](x^4 - 3x^2 + x) + (2x^4 + 4x - 7) = 3x^4 - 3x^2 + 5x - 7[/tex]

The third expression is simplified as follows:

(x² - 2x)(2x + 3) = 2x³ + 3x² - 4x² - 6x = 2x³ - x² - 6x.

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a. The t-value is 1.895.

b. The t-value is 1.734.

c. The t-value is 2.898.

To calculate the t-statistic for a confidence interval, we first need to determine the degrees of freedom (df), which depends on the sample size minus one. We can then use a t-distribution table, software, or calculator to find the t-value at the desired confidence level and degrees of freedom.

a. For a 90% confidence interval with 8 observations, the degrees of freedom is 7. Using the t-distribution table or calculator,

b. For a 90% confidence interval with 18 observations, the degrees of freedom is 17. Using the t-distribution table or calculator,

c. For a 99% confidence interval with 18 observations, the degrees of freedom is 17. Using the t-distribution table, software, or calculator,

Note that as the sample size increases, the degrees of freedom increase and the t-value approaches the value of the standard normal distribution for large sample sizes. This means that for large sample sizes, we can use the z-value instead of the t-value in confidence interval calculations.

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Answer:

Using the sine function, we have:

sin(angle) = opposite / hypotenuse

sin(22 degrees) = opposite / 18 feet

We can rearrange this equation to solve for the opposite side (height of the building):

opposite = sin(22 degrees) * 18 feet

Calculating this:

opposite ≈ 6.24 feet

Therefore, the base of the ladder from the base of the building is approximately 6.24 feet.

Answer:

x = 11.62

Step-by-step explanation:

opp / adj would be tangent, so the equation would be 4*tan(71) giving us 11.616, and rounding to the nearest hundredth gives you 11.62

Hope this helps :)

Answer:

Step-by-step explanation:

1. The system of inequalities that models this scenario can be represented as:

Let x be the number of servings of dry food.

Let y be the number of servings of wet food.

The cost constraint:

1x + 5y ≤ 15

The minimum number of dogs constraint:

x + y ≥ 4

2. The graph of the system of inequalities would be a shaded region in the coordinate plane.

To graph the inequality 1x + 5y ≤ 15, we can first graph the equation 1x + 5y = 15 (the corresponding boundary line) by finding two points on the line and connecting them. For example, when x = 0, y = 3, and when y = 0, x = 15. Plotting these points and drawing a line through them will represent the equation 1x + 5y = 15.

Next, we need to shade the region below the line because the inequality is less than or equal to (≤). This shaded region represents the solutions that satisfy the cost constraint.

To graph the inequality x + y ≥ 4, we can again find two points on the line x + y = 4 (the corresponding boundary line). For example, when x = 0, y = 4, and when y = 0, x = 4. Plotting these points and drawing a line through them will represent the equation x + y = 4.

Lastly, we shade the region above the line x + y = 4 because the inequality is greater than or equal to (≥). This shaded region represents the solutions that satisfy the minimum number of dogs constraint.

The solution set is the overlapping region where the shaded areas of both inequalities intersect. This region represents the combination of servings of dry food and wet food that Michelle can purchase within her budget ($15) to feed at least four dogs at the animal shelter.

The inequalities D + W > 4 and D + 5W ≤ 15 model the problem. The graph represents these inequalities, with the overlap of shaded regions showing possible food serving combinations.

Let's define D as the number of servings of dry food and W as the number of servings of wet food. The system of inequalities that models this scenario is:

The graph will show the solution sets to the inequalities. D and W must both be non-negative, hence the graphed area is in the first quadrant. The first inequality requires shading above a line that connects (0,4) and (4,0). This line is solid since numbers equal to 4 are included. The second inequality requires shading below a line that connects (0,3) and (15,0). This is also a solid line because Michelle can spend exactly $15. The overlapping region of the graph is the solution set, quantifying the combinations of dry and wet food servings that Michelle can buy.

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The initial stress is 1.273 × [tex]10^9[/tex] N/[tex]m^2[/tex], the resultant stress is 1.273 × [tex]10^9[/tex] N/[tex]m^2[/tex] and the induced force in the bar when the temperature reaches 50°C is 100.03 kN.

To calculate the initial stress in the tie bar, we can use the formula:

Stress = Load/Area

The area of the tie bar can be calculated using the formula for the area of a circle:

Area = π * [tex](diameter/2)^2[/tex]

Plugging in the values, we get:

Area = π * [tex]10 mm^{2}[/tex] = π *[tex](5 mm)^2[/tex] = 78.54 [tex]mm^2[/tex]

Converting the area to square meters, we have:

Area = 78.54 [tex]mm^2[/tex]* (1 m^2 / 1,000,000 [tex]mm^2[/tex]) = 7.854 × 1[tex]0^-5 m^2[/tex]

Now we can calculate the initial stress:

Initial Stress = 100 kN / 7.854 ×[tex]10^-5 m^2[/tex] = 1.273 × [tex]10^9 N/m^2[/tex]To calculate the resultant stress when the temperature rises to 50°C, we need to consider the thermal expansion of the tie bar. The change in length can be calculated using the formula:

ΔL = α * L0 * ΔT

Where ΔL is the change in length, α is the coefficient of linear expansion, L0 is the initial length, and ΔT is the change in temperature.

The induced force in the bar can be calculated using the formula:

Induced Force = Initial Stress * Area + E * α * ΔT * Area

Plugging in the values, we get:

Induced Force = (1.273 × 10^9 N[tex]m^2[/tex] * 7.854 × [tex]10^-5 m^2[/tex]) + (200 × [tex]10^9[/tex] N/[tex]m^2[/tex] * 14 × [tex]10^-6[/tex] /K * (50 - 15) K * 7.854 × [tex]10^-5 m^2[/tex])

Simplifying the equation, we find:

Induced Force = 100.03 kN

Therefore, the initial stress is 1.273 × [tex]10^9[/tex] N/[tex]m^2[/tex], the resultant stress is 1.273 × [tex]10^9[/tex] N/[tex]m^2[/tex], and the induced force in the bar when the temperature reaches 50°C is 100.03 kN.

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The probable question may be:

A rigidly held tie bar in a heating chamber has a diameter of 10 mm and is tensioned to a load of 100 kN at a temperature of 15°C. What is the initial stress, the resultant stress and what will be the induced force in the bar when the temperature in the chamber has risen to 50°C? E= 200 GN/ m2 and the coefficient of linear expansion of the material for tie bar = 14 × 10−6 /K.

Answer:

c) 54 feet

Step-by-step explanation:

Let the  side of the square be x

perimeter = 4x

⇒ 36 = 4x

⇒ x = 36/4 = 9 feet

Each side is 9 feet

When we join the two squares, it becomes a rectangle with

b = 9

l = 9 + 9 = 18

The perimeter of a rectangle is 2(l + b)

perimeter = 2(18 + 9)

= 2(27)

= 54 feet

Answer:

Option (c) 54 feet

Step-by-step explanation:

Perimeter of one square is 36 feet.

Side of square = 36 / 4 = 9 feet

When combined together one side of each square merged.

So, the perimeter of rectangular shape will be;

(9+9+9) + (9+9+9)

27 + 27

54 feet.

The loaded die appears to behave differently from a fair die.Based on the chi-square test, the loaded die does not exhibit equal probabilities for each outcome.

To test the claim that the outcomes of the loaded die are not equally likely, we can use the chi-square goodness-of-fit test. The null hypothesis (H₀) assumes that the outcomes are equally likely, while the alternative hypothesis (H₁) assumes that they are not equally likely.

Step 1: Set up hypotheses

H₀: The outcomes of the die are equally likely.

H₁: The outcomes of the die are not equally likely.

Step 2: Select the significance level

The significance level is given as 0.025, which means we have a two-tailed test and an alpha level of 0.025 for each tail.

Step 3: Compute the test statistic

We calculate the chi-square test statistic using the observed frequencies and the expected frequencies assuming equal probabilities for each outcome.

Expected frequencies (fair die):

1: 200/6 = 33.33

2: 200/6 = 33.33

3: 200/6 = 33.33

4: 200/6 = 33.33

5: 200/6 = 33.33

6: 200/6 = 33.33

Applying the chi-square formula, we get the test statistic:

χ² = ∑((observed - expected)² / expected)

Calculating this value, we get χ² ≈ 13.97.

Step 4: Determine the critical value

Since the significance level is 0.025 and the test is two-tailed, we divide the significance level by 2 to find the critical value associated with each tail. With 5 degrees of freedom (6 categories - 1), the critical value is approximately 11.07.

Step 5: Make a decision

The test statistic (13.97) is greater than the critical value (11.07), which leads us to reject the null hypothesis. Thus, we have evidence to suggest that the outcomes of the loaded die are not equally likely.

The loaded die appears to behave differently from a fair die.

In conclusion, based on the chi-square test, the loaded die does not exhibit equal probabilities for each outcome.

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Answer:

20 dimes

Step-by-step explanation:

1 dime = $0.1

1 quarter = $0.25

Let there be x dimes

Then the number of quarters is 32-x

Also, $5 = 0.1x + 0.25(32-x)

⇒ 5 = 0.1x + 8 - 0.25x

⇒ 0.25x - 0.1x = 8 - 5

⇒ 0.15x = 3

⇒ x = 3/0.15

⇒ x = 20

To show that ΔJKL ≅ ΔMNR by SAS (Side-Angle-Side), we need the additional information that the lengths of the corresponding sides JK and MN are equal.

To prove ΔJKL ≅ ΔMNR using the SAS congruence criterion, we need to establish that two corresponding sides and the included angle of the triangles are congruent.

1. Given information:

  - KL ≅ NR (corresponding sides)

  - JL ≅ MR (corresponding sides)

  - ∠J ≅ ∠M (included angle)

  - ∠L ≅ ∠R (corresponding angles)

  - ∠K ≅ ∠N (corresponding angles)

  - ∠R ≅ ∠K (corresponding angles)

2. Additional information needed:

  - We need to know if JK ≅ MN (corresponding sides) to establish the SAS congruence criterion.

3. Possible scenarios:

  - If JK ≅ MN, then we can establish that ΔJKL ≅ ΔMNR by SAS.

  - If JK is not equal to MN, then we cannot apply the SAS congruence criterion, and additional information or a different congruence criterion would be needed to prove the triangles congruent.

In summary, the lengths of the corresponding sides JK and MN need to be equal to prove ΔJKL ≅ ΔMNR by SAS. Without this information, we cannot conclude the congruence of the triangles using the SAS criterion alone.

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The value of m<RQS as required to be determined in the task content is; 70°.

It follows from the task content that the measure of angle RQS is to be determined as required.

Recall, the measure of the central angle subtended by an arc is twice that which it subtends at any point on the circumference.

Therefore, m<RPS = 2 • m<RQS.

m<RQS = 140°/2

m<RQS = 70°.

Ultimately, the measure of angle RQS are; 70°.

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Answer:

-2097152 is the 20th term

Step-by-step explanation:

Write geometric sequence as an explicit formula

[tex]-4,8,-16,32\rightarrow-4(-2)^0,-4(-2)^1,-4(-2)^2,4(-2)^3\\a_n=a_1r^{n-1}\rightarrow a_n=-4(-2)^{n-1}[/tex]

Find the n=20th term

[tex]a_{20}=-4(-2)^{20-1}=-4(-2)^{19}=4(-524288)=-2097152[/tex]

Answer:

5n - 8 = 37

5n = 45

n = 9

The number is 9.

Given the information in the diagram, the theorem that best justifies why lines j and k must be parallel include the following: D. converse alternate exterior angles theorem.

In Mathematics and Geometry, parallel lines are two (2) lines that are always the same (equal) distance apart and never meet or intersect.

In Mathematics and Geometry, the alternate exterior angle theorem states that when two (2) parallel lines are cut through by a transversal, the alternate exterior angles that are formed lie outside the two (2) parallel lines, are located on opposite sides of the transversal, and are congruent angles.

Since the alternate exterior angles are congruent, we can logically deduce the following based on the converse alternate exterior angles theorem;

93° ≅ 93° (lines j and k are parallel lines).

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Complete Question:

Given the information in the diagram, which theorem best justifies why lines j and k must be parallel?

alternate interior angles theorem

alternate exterior angles theorem

converse alternate interior angles theorem

converse alternate exterior angles theorem

49.87 because I searched it up

The equation of the new function after shifting 6 units right and 4 units down is f(x) = (x + 6)² - 4.

If we are to apply the changes below to the quadratic parent function, F(x) = x², what is the equation of the new function, given that we are to shift 6 units to the right and 4 units down? We will approach this question by following the steps outlined below.

Step 1: Identify the parent function F(x) = x² and its transformations

Step 2: Write the equation of the new function

Step 3: Simplify the new equation of the function.Step 1: Identify the parent function F(x) = x² and its transformations

Here, we are given the quadratic parent function F(x) = x² and two transformations: shift 6 units right and shift 4 units down.

The general equation for the horizontal and vertical shifts of a quadratic function is given by:f(x) = a(x - h)² + k, where a, h, and k are constants.

The value of a determines the direction of opening of the parabola, while (h, k) represents the vertex of the parabola.

If a > 0, the parabola opens upwards, while a < 0, the parabola opens downwards. If the values of (h, k) are positive, the parabola is shifted right and up, respectively. On the other hand, if the values of (h, k) are negative, the parabola is shifted left and down, respectively.

Therefore, given the quadratic parent function F(x) = x² and two transformations: shift 6 units right and shift 4 units down, we can represent these changes by the following:

a = 1 (since the parabola opens upwards)h = -6 (since we are shifting the parabola 6 units to the right)k = -4 (since we are shifting the parabola 4 units down)

Step 2: Write the equation of the new function Now that we have identified the constants a, h, and k, we can write the equation of the new function as follows:f(x) = a(x - h)² + kf(x) = 1(x - (-6))² + (-4)Replacing the constants a, h, and k in the equation, we have:f(x) = (x + 6)² - 4

Step 3: Simplify the new equation of the function.f(x) = (x + 6)² - 4= (x + 6)(x + 6) - 4= x² + 12x + 36 - 4= x² + 12x + 32Therefore, the equation of the new function after shifting 6 units right and 4 units down is f(x) = x² + 12x + 32.

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The point of intersection is (-1, 0). Therefore, the correct answer is C. (-1, 0).

To find the solution to the system of equations graphed below, we need to identify the point where the two lines intersect.

The equations of the lines are:

y = -x - 1

y = x + 1

To find the point of intersection, we can set the two equations equal to each other and solve for x:

-x - 1 = x + 1

Adding x to both sides:

-1 = 2x + 1

Subtracting 1 from both sides:

-2 = 2x

Dividing both sides by 2:

x = -1

Now that we have the x-coordinate, we can substitute this value back into either equation to find the y-coordinate. Let's use the second equation:

y = x + 1

y = -1 + 1

y = 0

Therefore, the point of intersection is (-1, 0).

Among the given options, the closest point to (-1, 0) is option C: (-1, 0).

Therefore, the correct answer is C. (-1, 0).

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The correct statements are: B. Circle F and circle D are similar and D. The center of circle F is (0, 3).

Explanation:

A. The radius of circle F is not 28. When a circle is translated, the radius remains the same. So, circle F has the same radius as circle D, which is 7.

B. Circle F and circle D are similar. Similarity means that the two shapes have the same shape but possibly different sizes. Since circle F is a translation of circle D, it has the same shape and proportions as circle D. Therefore, they are similar.

C. Circle F and circle D are not congruent. Congruence means that two shapes are identical in both shape and size. While circle F and circle D have the same shape, they have different positions due to the translation. Thus, they are not congruent.

D. The center of circle F is (0, 3). When a circle is translated horizontally by a certain amount, the x-coordinate of the center changes, while the y-coordinate remains the same. Since circle F is translated 2 units to the right, the x-coordinate of the center of circle F would be 2 + 0 = 2. As the y-coordinate remains the same, the center of circle F is (2, 3). Therefore, the statement is incorrect.

In summary, the correct statements are B and D. Circle F and circle D are similar, and the center of circle F is (0, 3). Therefore, Option B and D are correct.

The question was incomplete. find the full content below:

According to these three facts, which statements are true?

• Circle D has center (2, 3) and radius 7.

• Circle F is a translation of circle D, 2 units right.

• Circle F is a dilation of circle D with a scale factor of 4.

Select each correct answer.

A. The radius of circle F is 28.

B. Circle F and circle D are similar.

C. Circle F and circle D are congruent.

D. The center of circle F is (0, 3)

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The predicted sales in month 5 is -2778.

Obtaining the linear equation which models the data :

b = slope = (y2-y1)/(x2-x1)

b = (926-7408)/(4-1)

b = -2160.67

c = intercept ;

taking the points (x = 2 and y = 3704)

Inserting into the general equation:

3704 = -2160.67(2) + c

3704 = -4321.33 + c

c = 3704 + 4321.33

c = 8025.33

General equation becomes : y = -2160.67x + 8025.33

To obtain sales in month 5:

y = -2160.67(5) + 8025.33

y = -2778

Hence, the predicted sales in month 5 is -2778.

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Answer:

See below

Step-by-step explanation:

To solve for "a", you would first isolate ab on the right-hand side by adding "c" on both sides:

[tex]ab-c=d\\ab-c+c=d+c\\ab=d+c[/tex]

Next, you would divide both sides by "b" to isolate "a":

[tex]ab=d+c\\ab\div b=(d+c)\div b\\a = \frac{d+c}{b}[/tex]

The solution is:

Work/explanation:

To solve the given expression for a, we should isolate it by using basic algebraic operations.

The first step is to add c to each side:

[tex]\sf{ab-c=d}[/tex]

[tex]\sf{ab=d+c}[/tex]

Now, divide each side by b:

[tex]\sf{a=\dfrac{d+c}{b}}[/tex]

I have solved the equation for a.

The insurance policy would cover up to $216,000 for the item or property.

If the replacement value of an item or property is $270,000 and your insurance policy provides coverage for 80% of the replacement value, it means that your insurance policy will cover up to 80% of the $270,000 in the event of a covered loss.

To calculate the coverage amount, you would multiply the replacement value by the coverage percentage:

Coverage Amount = Replacement Value * Coverage Percentage

In this case:

Coverage Amount = $270,000 * 0.80

Coverage Amount = $216,000

Therefore, your insurance policy would cover up to $216,000 for the item or property in question. Any losses exceeding this coverage amount would not be covered by your insurance policy.

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One of the transformations undergone by Triangle 1 is a rotation, which involves turning the triangle around a fixed point while preserving its shape and size.

A rotation is a transformation that turns an object around a fixed point, known as the center of rotation. In the given results, if the triangle appears in a different orientation but retains its shape and size, it indicates a rotation.

During a rotation, each point of the triangle is moved along a circular path around the center of rotation. The distance from the center of rotation remains constant, and the angle between any two corresponding points on the original and rotated triangles is preserved. The direction of rotation can be clockwise or counterclockwise, depending on the given results.

To describe a rotation, we need to specify the angle of rotation and the direction. For example, "Triangle 1 underwent a counterclockwise rotation of 90 degrees" would indicate that the triangle was rotated by 90 degrees in the counterclockwise direction.

The specific rotation can be described by stating the angle of rotation and the direction.

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