For a circle of radius 4 feet, find the arc length s subtended by a central angle of 26 degrees.

Applying implicit differentiation, it is found that:

[tex]\frac{d^2y}{dx^2}(4,3) = \frac{7}{27}[/tex]

The curve given is:

[tex]x^2 + y^2 = 25[/tex]

Applying implicit differentiation, we have that the first derivative is:

[tex]2x\frac{dx}{dx} + 2y\frac{dy}{dx} = 0[/tex]

[tex]2y\frac{dy}{dx} = -2x[/tex]

[tex]\frac{dy}{dx} = -\frac{x}{y}[/tex]

Again applying implicit differentiation, the second derivative is:

[tex]\frac{d^2y}{dx^2} = -\frac{y\frac{dx}{dx} - x\frac{dy}{dx}}{y^2}[/tex]

Since [tex]\frac{dy}{dx} = -\frac{x}{y}[/tex]

[tex]\frac{d^2y}{dx^2} = -\frac{y - \frac{x^2}{y}}{y^2}[/tex]

[tex]\frac{d^2y}{dx^2} = \frac{x^2 - y^2}{y^3}[/tex]

At point (4,3), [tex]x = 4, y = 3[/tex], then:

[tex]\frac{d^2y}{dx^2}(4,3) = \frac{4^2 - 3^2}{3^3} = \frac{7}{27}[/tex]

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Final answer:

To simplify the expression sec(y) sin(π/2 - y) using basic identities, we can use the identity sec(y) = 1/cos(y) and expand the expression using the identity sin(a - b) = sin(a)cos(b) - cos(a)sin(b). The simplified expression is -tan(y).

Explanation:

To simplify the expression sec(y) sin(π/2 - y) using basic identities, we can start by using the identity sec(y) = 1/cos(y). So the expression becomes (1/cos(y)) sin(π/2 - y). Next, let's use the identity sin(a - b) = sin(a)cos(b) - cos(a)sin(b) to expand the expression. We have (1/cos(y)) (cos(y)cos(π/2) - sin(y)sin(π/2)). Since cos(π/2) = 0 and sin(π/2) = 1, the expression simplifies to (1/cos(y)) (0 - sin(y)(1)), which further simplifies to -sin(y)/cos(y). Therefore, the simplified expression is -tan(y).

Answer:

C The pre-image will be congruent to the image.

Step-by-step explanation:

Final answer:

Given that A is a subset of B and the intersection of A and B is the empty set, it is conclusive that A is the empty set, and we cannot infer anything about the elements of set B.

Explanation:

When considering sets, subset relation and intersection are fundamental concepts. If A ⊆ B(A is a subset of B) and A ∩ B = θ (the intersection of A and B is the empty set), we can conclude that every element in A is also in B, but since the intersection is empty, A has no elements. Therefore, A is the empty set, which means that A contains no elements. It is not possible to determine the contents of set B or to compare the cardinalities of A and B, since we only know that A is empty.

Observing our subset and intersection principles, while A is a subset of B, the fact that they have an empty intersection invariably means that A cannot contain any member, thereby being the empty set. The other options given in the question are incorrect based on our set theory principles. Set B can be any set; it might not be empty, and it is not possible to make any conclusions about the number of elements in B relative to A from the information given.

Answer:

D is the answer

Step-by-step explanation:

The ratios aren't the same, the ratios must be the same.

Final answer:

The probability of rolling a number less than or equal to 8 with two dice, given at least one die shows a 6, is 5/36, considering all the valid dice combinations that meet the criteria.

Explanation:

The probability of rolling a number less than or equal to 8 with two dice, given at least one must be a 6, is calculated by considering the possible combinations of dice that adhere to these conditions. First, we determine how many outcomes include at least one 6:

If the first die is a 6, the other die can be 1, 2, or a 3 (since 6+4 is already 10, which is greater than 8).

If the second die is a 6, the first die can again be 1, 2, or 3.

Both dice can be 6, which adds one more outcome.

These are mutually exclusive events, therefore we add their probabilities. There are 5 outs of 36 possible rolls (3+3) when the numbers can sum up to less than or equal to 8 with at least one die showing a 6 (1+6, 2+6, 3+6, 6+1, 6+2, 6+3).

The probability of rolling a number less than or equal to 8, given that at least one die shows a 6, is 5/36.

The probability of rolling a number less than or equal to 8 with two dice, given that at least one die shows a 6, is [tex]\( \frac{10}{11} \).[/tex]

To find the probability of rolling a number less than or equal to 8 with two dice, given that at least one of the dice must show a 6, we can use conditional probability.

Let's consider the outcomes where at least one die shows a 6:

- (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)

- (1, 6), (2, 6), (3, 6), (4, 6), (5, 6)

Out of these 11 outcomes, the ones where the sum of the two dice is less than or equal to 8 are:

- (6, 1), (6, 2), (6, 3), (6, 4), (6, 5)

- (1, 6), (2, 6), (3, 6), (4, 6), (5, 6)

So, there are 10 favorable outcomes out of 11 possible outcomes when at least one die shows a 6.

Therefore, the probability of rolling a number less than or equal to 8 with two dice, given that at least one of the dice must show a 6, is [tex]\( \frac{10}{11} \).[/tex]

The answer is option C "multiplication." You have the function p(x) = 30x + 2 and x equals the number of years since the company started producing games which has this equation also s(x) = 500x + 250 in order to find the revenue from their video games the company should use the multiplication on the polynomials.

Hope this helps!

Final answer:

The third piece of wood must be longer than 9 ft but shorter than 35 ft to make a triangular garden bed with the other two pieces, adhering to the Triangle Inequality Theorem.

Explanation:

To determine the possible length of the third piece of wood to create a triangular garden bed with the other two pieces measuring 22 ft and 13 ft, we need to recall the Triangle Inequality Theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. Conversely, the difference between these two sides must be less than the length of the remaining side.

Step 1: Identify the lengths of the two given pieces: 22 ft and 13 ft.

Step 2: To find the minimum length of the third piece, subtract the shorter piece from the longer piece, then add a small value to avoid equality (as the sum must be greater).

Minimum length > 22 ft - 13 ft => 9 ft + a small value

Step 3: To find the maximum length of the third piece, sum the lengths of the two pieces and subtract a small value (as the length must be less than the sum of the other two pieces).

Maximum length < 22 ft + 13 ft => 35 ft - a small value

So, the third piece of wood must be longer than 9 ft but shorter than 35 ft to form a triangle with the other two pieces.

The total cost of a movie ticket for club members is $20 ($15 membership fee + $5 ticket price) and for non-members it is $8.

To find the total cost of a movie ticket for club members and non-members, we need to consider the membership fee and the ticket price. Let's break it down:

For club members:

For non-members:

To calculate the total cost for club members, we add the membership fee to the ticket price: $15 + $5 = $20.

To find the total cost for non-members, we simply use the ticket price: $8.

The system of equations representing the total cost for a club member (y) and a nonmember (y) after viewing x movies is given by:

y = 15 + 5x

y = 8x

Let's break down the system of equations. For a club member, the total cost (y) is the sum of the membership fee ($15) and the product of the ticket price $5 and the number of movies viewed (x). This relationship is represented by the equation y = 15 + 5x.

On the other hand, for a nonmember, the total cost (y) is solely the product of the ticket price for nonmembers $8 and the number of movies viewed (x), as there is no membership fee. This is expressed by the equation y = 8x.

To find the number of movies (x) after which the total cost (y) for a club member equals the cost for a nonmember, we set the two equations equal to each other:

15 + 5x = 8x

Now, solving for x, we can determine the point at which the total cost for a club member is the same as the cost for a nonmember. This helps in understanding the break-even point where joining the movie club becomes financially advantageous, factoring in the membership fee and the reduced ticket price for club members.

The complete question of the above answer is:

Certainly, here is the complete question:

It costs $15 for a yearly membership to a movie club at a movie theater. A movie ticket costs $5 for club members and $8 for nonmembers.

a. Write a system of equations that you can use to find the number x of movies viewed after which the total cost y for a club member, including the membership fee, is the same as the cost for a nonmember.

To determine the time it takes for the computer to sort 10 objects, substitute x with 10 in the given function, resulting in 0.52 seconds, rounded to the nearest hundredth.

The student's question involves determining how long it takes a computer to sort 10 objects, given the function t = 0.005x² + 0.002x, where x is the number of objects and t is the time in seconds. To find out the time it takes to sort 10 objects, we substitute x with 10 into the function.

So, we have:

t = 0.005(10)² + 0.002(10)

t = 0.005(100) + 0.002(10)

t = 0.50 + 0.02

t = 0.52 seconds

Therefore, it takes the computer approximately 0.52 seconds to sort 10 objects, when the result is rounded to the nearest hundredth of a second.

Answer:

The answer would be 123.4

Step-by-step explanation:

Resiprical

The number 4.24 in its decimal form is already a simplified numerical value, it does not require conversion to a simplified radical form, unless it's interpreted as the square root of 4.24.

The student is wanting to know the simplified radical form of 4.24. The student's proposed answer of 106/25 indicates that they are interpreting the number as a fraction. As the number 4.24 is already a decimal, which is a form of a simplified numerical expression, it does not need conversion to a simplified radical form or fraction. A number in radical form typically denotes the square root of a number, for which 4.24 does not have a simple whole number equivalent.

In summary, 4.24 in its decimal expression is already a simplified numerical value. Unless it is interpreted as the square root of 4.24, which represents an entirely different value and is not commonly presented in radical form when it's a decimal.

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The number 4.24 can be simplified to 106/25.

No, actually, you're right!  The decimal number 4.24 can be expressed as the simplified fraction 106/25.

Here is how:

1. First, express the number 4.24 as a fraction, understanding that the number after the decimal point stands for hundredths. Thus, we can write it as 424/100.

2. Next, we need to simplify the fraction. To do that, we find the greatest common divisor (GCD) of 424 and 100. The gcd of 424 and 100 is 4.

3. Now we divide both the numerator and the denominator by the gcd. So, 424 divided by 4 is 106, and 100 divided by 4 is 25.

4. Now we write the simplified fraction as 106/25.

However, it seems there's a misunderstanding in the question. The term 'radical form' usually refers to the square root of a number which we can only do in contrast to perfect square numbers (those whose square root is a whole number). However, the decimal number 4.24 doesn't have a perfect square root, so it can't be expressed in 'radical form'. But, as per your query, 106/25 is the simplified fractional form of 4.24.

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Final answer:

To write 90 as a product of prime factors, divide it by the smallest prime numbers in sequence: 90 = 2 × 3 × 3 × 5, which can also be expressed using exponents as 90 = 2 × 3² × 5.

Explanation:

Writing 90 as a product of prime factors involves breaking down the number into its prime number components. To do this, we start by finding the smallest prime number that divides into 90, which is 2. We then divide 90 by 2 to get 45. Since 45 is not a prime number, we continue the process and find the smallest prime number that divides into 45, which is 3. Dividing 45 by 3 gives us 15. Again, 15 is not prime, so we divide it by its smallest prime divisor, 3, which results in 5. Now, 5 is a prime number, so we have now expressed 90 as a product of prime factors:

90 = 2 × 3 × 3 × 5

Step-by-Step Process:

Divide 90 by the smallest prime number 2, resulting in 45.

Divide 45 by the smallest prime number 3, resulting in 15.

Divide 15 by 3 again, resulting in 5.

Write down the result as a multiplication of these prime factors: 2 × 3 × 3 × 5.

This factorization can also be written with exponents to show repeated factors: 90 = 2 × 3² × 5.

Answer:

D

Step-by-step explanation:

She showed that f(n) - f(n - 1) was a constant difference.

The definition of an arithmetic sequence is the difference between consecutive terms is constant, aka common difference.

Answer: the correct answer is D.

Step-by-step explanation:

the one above is incorrect the real answer is 10 meters per second and you don't really need an equation for this (it's pretty obvious ) :)