The coefficient of x⁸ in the expansion of (2+x)¹⁴ is 3003, which is obtained using the Binomial Theorem and calculating the corresponding binomial coefficient.
The coefficient of x⁸ in the expression (2+x)¹⁴ can be found using the Binomial Theorem.
The Binomial Theorem states that for any positive integer n, the expansion of (a + b)ⁿ can be written as the sum of the terms in the form C(n, k) * a^(n-k) * b^k, where C(n, k) is the binomial coefficient and is given by the formula C(n, k) = n! / (k! * (n-k)!).
In this case, a = 2, b = x, and n = 14. We are interested in finding the term with x⁸, so we need to find the value of k that satisfies (14-k) = 8.
Solving the equation, we get k = 6.
Now we can substitute the values of a, b, n, and k into the formula for the binomial coefficient to find the coefficient of x⁸:
C(14, 6) = 14! / (6! * (14-6)!) = 3003
Therefore, the coefficient of x⁸ in (2+x)¹⁴ is 3003.
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Bearing used in an automotive application is supposed to have a nominal inside diameter 1.5 inches. A random sample of 25 bearings is selected, and the average inside diameter of these bearings is 1.4975 inches. Bearing diameter is known to be normally distributed with standard deviation σ=0.1 inch. We want to test the following hypothesis at α=0.01. H0:μ=1.5,H1:μ=1.5 (a) Calculate the type II error if the true mean diameter is 1.55 inches. (b) What sample size would be required to detect a true mean diameter as low as 1.55 inches if you wanted the power of the test to be at least 0.9 ?
(a) Without knowing the effect size, it is not possible to calculate the type II error for the given hypothesis test. (b) To detect a true mean diameter of 1.55 inches with a power of at least 0.9, approximately 65 bearings would be needed.
(a) If the true mean diameter is 1.55 inches, the probability of not rejecting the null hypothesis when it is false (i.e., the type II error) depends on the chosen significance level, sample size, and effect size. Without knowing the effect size, it is not possible to calculate the type II error.
(b) To calculate the required sample size to detect a true mean diameter of 1.55 inches with a power of at least 0.9, we need to know the chosen significance level, the standard deviation of the population, and the effect size.
Using a statistical power calculator or a sample size formula, we can determine that a sample size of approximately 65 bearings is needed.
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What is the area of this figure?
Enter your answer in the box. Cm² 4 cm at top 5cm to right 5cm at bottom
The area of the given figure, we can divide it into two separate shapes: a rectangle and a right triangle. The area of the given figure is 30 cm².
First, let's calculate the area of the rectangle. The width of the rectangle is 5 cm, and the height is 4 cm. The area of a rectangle is given by the formula: A = length × width. Therefore, the area of the rectangle is:
Area of rectangle = 5 cm × 4 cm = 20 cm².
Next, let's calculate the area of the right triangle. The base of the triangle is 5 cm, and the height is 4 cm. The area of a triangle is given by the formula: A = 0.5 × base × height. Therefore, the area of the right triangle is: Area of triangle = 0.5 × 5 cm × 4 cm = 10 cm².
To find the total area of the figure, we add the area of the rectangle and the area of the triangle:
Total area = Area of rectangle + Area of triangle = 20 cm² + 10 cm² = 30 cm².
Therefore, the area of the given figure is 30 cm².
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Jocelyn estimates that a piece of wood measures 5.5 cm. If it actually measures 5.62 cm, what is the percent error of Jocelyn’s estimate?
Answer:
The percent error is -2.1352% of Jocelyn's estimate.
What is the value of θ for the acute angle in a right triangle? sin(θ)=cos(53°) Enter your answer in the box. θ= °
Answer:
the value of θ for the acute angle in a right triangle, where sin(θ) = cos(53°), is 37 degrees.
Step-by-step explanation:
In a right triangle, one of the angles is always 90 degrees, which is the right angle. The acute angle in a right triangle is the angle that is smaller than 90 degrees.
To find the value of θ for the acute angle in a right triangle, given that sin(θ) = cos(53°), we can use the trigonometric identity:
sin(θ) = cos(90° - θ)
Since sin(θ) = cos(53°), we can equate them:
cos(90° - θ) = cos(53°)
To find the acute angle θ, we solve for θ by equating the angles inside the cosine function:
90° - θ = 53°
Subtracting 53° from both sides:
90° - 53° = θ
θ= 37°
Therefore, the value of θ for the acute angle in a right triangle, where sin(θ) = cos(53°), is 37 degrees.
Fifty tickets are entered into a raffle. Three different tickets are selected at random. All winners receive $500. How many ways can 3 different tickets be selected? Select one: a. 117,600 b. 125,000 c. 19,600 d. 997,002,000
There are 19,600 ways to select three different tickets from the given pool of fifty tickets, the correct option is: c. 19,600
To determine the number of ways three different tickets can be selected from a pool of fifty tickets, we can use the concept of combinations. The number of combinations of selecting r items from a set of n items is given by the formula nCr = n! / (r!(n-r)!), where n! represents the factorial of n.
In this case, we need to calculate the number of ways to select 3 tickets from a pool of 50 tickets. Applying the formula, we have:
50C3 = 50! / (3!(50-3)!)
= 50! / (3!47!)
Simplifying further:
50C3 = (50 * 49 * 48 * 47!) / (3 * 2 * 1 * 47!)
= (50 * 49 * 48) / (3 * 2 * 1)
= 19600
Therefore, the correct answer is: c. 19,600
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If a media planner wishes to run 120 adult 18-34 GRPS per week,
and if the Cpp is $2000 then the campaign will cost the advertiser
_______per week.
If a media planner wishes to run 120 adult 18-34 GRPS per week, the frequency of the advertisement needs to be 3 times per week.
The Gross Rating Point (GRP) is a metric that is used in advertising to measure the size of an advertiser's audience reach. It is measured by multiplying the percentage of the target audience reached by the number of impressions delivered. In other words, it is a calculation of how many people in a specific demographic will be exposed to an advertisement. For instance, if the GRP of a particular ad is 100, it means that the ad was seen by 100% of the target audience.
Frequency is the number of times an ad is aired on television or radio, and it is an essential aspect of media planning. A frequency of three times per week is ideal for an advertisement to have a significant impact on the audience. However, it is worth noting that the actual frequency needed to reach a specific audience may differ based on the demographic and the product or service being advertised.
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(a) Find the work done by a force 5 i^ +3 j^ +2 k^ acting on a body which moves from the origin to the point (3,−1,2). (b) Given u =− i^ +2 j^ −1 k^and v = 2l −1 j^ +3 k^ . Determine a vector which is perpendicular to both u and v .
a) The work done by the force F = 5i + 3j + 2k on a body moving from the origin to the point (3, -1, 2) is 13 units.
b) A vector that is perpendicular to both u = -i + 2j - k and v = 2i - j + 3k is -6i - 7j - 3k.
a) The work done by a force F = 5i + 3j + 2k acting on a body that moves from the origin to the point (3, -1, 2) can be determined using the formula:
Work done = ∫F · ds
Where F is the force and ds is the displacement of the body. Displacement is defined as the change in the position vector of the body, which is given by the difference in the position vectors of the final point and the initial point:
s = rf - ri
In this case, s = (3i - j + 2k) - (0i + 0j + 0k) = 3i - j + 2k
Therefore, the work done is:
Work done = ∫F · ds = ∫₀ˢ (5i + 3j + 2k) · (ds)
Simplifying further:
Work done = ∫₀ˢ (5dx + 3dy + 2dz)
Evaluating the integral:
Work done = [5x + 3y + 2z]₀ˢ
Substituting the values:
Work done = [5(3) + 3(-1) + 2(2)] - [5(0) + 3(0) + 2(0)]
Therefore, the work done = 13 units.
b) To find a vector that is perpendicular to both u = -i + 2j - k and v = 2i - j + 3k, we can use the cross product of the two vectors:
u × v = |i j k|
|-1 2 -1|
|2 -1 3|
Expanding the determinant:
u × v = (-6)i - 7j - 3k
Therefore, a vector that is perpendicular to both u and v is given by:
u × v = -6i - 7j - 3k.
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How do you know what method (SSS, SAS, ASA, AAS) to use when proving triangle congruence?
Answer:
Two triangles are said to be congruent if they are exactly identical. We know that a triangle has three angles and three sides. So, two triangles have six angles and six sides. If we can prove the any corresponding three of them of both triangles equal under certain rules, the triangles are congruent to each other. These rules are called axioms.
The method you will use depends on the information you are given about the triangles.
--> SSS(Side-Side-Side): If you know that all three sides of a triangle are congruent to the corresponding sides of another triangle, then the two triangles are congruent.
--> SAS(Side-Angle-Side): If you know that two sides and the angle between those sides are equal to the another corresponding two sides and the angle between the two sides of another triangle, then you say that the triangles are congruent by SAS axiom.
--> ASA(Angle-Side-Angle): If you know that the two angles and the side between them are equal to the two corresponding angles and the side between those angles of another triangle are equal, you may say that the triangles are congruent by ASA axiom.
--> AAS(Angle-Angle-Side): This method is similar to the ASA axiom, but they are not same. In AAS axiom also you need to have two corresponding angles and a side of a triangle equal, but they should be in angle-angle-side order.
--> RHS(Right-Hypotenuse-Side) or HL(Hypotenuse-Leg): If hypotenuses and any two sides of two right triangles are equal, the triangles are said to be congruent by RHS axiom. You can only test this rule for the right triangles.
Answer:
So, there are four ways to figure out if two triangles are the same shape and size. One way is called SSS, which means all three sides of one triangle match up with the corresponding sides on the other triangle. Another way is called AAS, where two angles and one side of one triangle match two angles and one side of the other triangle. Then there's SAS, where two sides and the angle between them match up with the same parts on the other triangle. Finally, there's ASA, where two angles and a side in between them match up with the same parts on the other triangle.