what is the least refractive index that a right-angle prism must have to reflect light through an angle of 90 degrees?

To find \(dy/dx\) by implicit differentiation, we will differentiate both sides of the equation \(5x^2 + 2y^3 = 10\) with respect to x.

Differentiating the left side:

\(d/dx(5x^2 + 2y^3) = d/dx(10)\)

Using the power rule, the derivative of \(x^2\) with respect to x is 2x:

\(10x + d/dx(2y^3) = 0\)

Now, we need to find \(d/dx(2y^3)\). To do this, we use the chain rule, which states that if we have a function of a function, the derivative is the derivative of the outer function multiplied by the derivative of the inner function.

For \(y^3\), the outer function is the cube function \(f(x) = x^3\) and the inner function is y(x).

\(d/dx(2y^3) = d/dx(2(f(y))^3) = 3(2(f(y))^2 * d/dx(f(y))\)

Using the chain rule again, we have:

\(= 3(2(f(y))^2 * f'(y) * dy/dx\)

Since \(f(y) = y\), the derivative of f(y) with respect to y is 1. Therefore,\(f'(y) = 1.\)

Substituting these values back into the equation, we have:

\(10x + 3(2(y)^2 * 1 * dy/dx = 0\)

Simplifying further:

\(10x + 6y^2 * dy/dx = 0\)

To isolate \(dy/dx\), we can subtract 10x from both sides:

\(6y^2 * dy/dx = -10x\)

Finally, we divide both sides by \(6y^2\) to solve for \(dy/dx\):

\(dy/dx = -10x / 6y^2\)

So, the derivative \(dy/dx\) for the equation \(5x^2 + 2y^3 = 10\), obtained by implicit differentiation, is \(-10x / 6y^2\).

Learn more about implicit differentiation here:

https://brainly.com/question/11887805

#SPJ4

The solutions to the Quadratic equation x² + 5x - 24 = 0 are x = 3 and x = -8.

In order to solve the given equation, it is essential to understand that there are three main steps to solve the quadratic equation, and these steps are as follows:

Step 1: Rearrange the terms and set them equal to zero.

Step 2: Factor the quadratic expression if possible or use the quadratic formula.Step 3: Solve for x by simplifying and evaluating the resulting expression. Now, let's apply these steps to the given quadratic equation, which is as follows: x² + 5x - 24 = 0

Step 1: Rearrange the terms and set them equal to zero the given quadratic equation is in standard form, which means the quadratic term (x²) is first, followed by the linear term (5x), and the constant term (-24) is on the right side. Thus, we can leave the equation as it is, because it is already set equal to zero.

Step 2: Factor the quadratic expression or use the quadratic formulaIn this case, the quadratic expression cannot be factored using integer values, so we must use the quadratic formula. The quadratic formula is as follows:$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$Where a, b, and c are the coefficients of the quadratic expression ax² + bx + c.

Therefore, we can identify the coefficients from the given equation as follows:a = 1, b = 5, c = -24.Now, we can substitute these values into the quadratic formula and solve for x as follows:$$x=\frac{-5\pm\sqrt{5^2-4(1)(-24)}}{2(1)}$$$$x=\frac{-5\pm\sqrt{25+96}}{2}$$$$x=\frac{-5\pm\sqrt{121}}{2}$$Step 3: Solve for x by simplifying and evaluating the resulting expression

Now, we can simplify the expression under the square root sign (the discriminant), which is 121, so we can rewrite the expression as follows:$$x=\frac{-5\pm\sqrt{121}}{2}$$$$x=\frac{-5\pm11}{2}$$$$x_1=\frac{-5+11}{2}=3$$$$x_2=\frac{-5-11}{2}=-8$$

Thus, the solutions to the quadratic equation x² + 5x - 24 = 0 are x = 3 and x = -8.

For more questions on Quadratic .

https://brainly.com/question/1214333

#SPJ8

I recommend using a class interval of 36.9 for this set of data.

To determine the recommended class interval for the given data, we can use the formula:

Class Interval = (Highest Value - Lowest Value) / Number of Classes

The number of classes is subjective, but a common choice is to use the Sturges' Rule, which is given by the formula:

Number of Classes = 1 + 3.3 * log10(Number of Observations)

Applying Sturges' Rule to the given data:

Number of Classes = 1 + 3.3 * log10(230)
Number of Classes ≈ 8.6

Round the number of classes up to the nearest whole number:

Number of Classes = 9

Now we can calculate the class interval:

Class Interval = (567 - 235) / 9
Class Interval = 332 / 9
Class Interval ≈ 36.9

Round the class interval to 1 decimal place:

Class Interval = 36.9

I recommend using a class interval of 36.9 for this set of data.

To know more about Class Interval, refer here:

https://brainly.com/question/19473137

#SPJ11

Answer:

$11.40

Step-by-step explanation:

$1.90*6=$11.40

This equation tells us that if we know the number of views in the second week (x), we can calculate the number of views in the third week by multiplying by 0.95.

0.95x = (number of views in the third week)

However, since we don't know the number of views in the second week, we can't calculate the exact number of views in the third week.

Let the number of views in the second week be x.

Then the number of views in the third week will be (0.95x), since the views decreased by 5%.

Therefore, if we know the number of views in the second week, we can calculate the number of views in the third week using the formula (0.95x).

Now, we are given that in the third week the views decreased by 5% from the second week. This means that the number of views in the third week was 95% of the number of views in the second week. We can write this as:

0.95x = (number of views in the third week)

Now, we don't know what the number of views in the second week (x) was.

However, we do know that in the third week, the views decreased by 5%. This implies that the number of views in the third week was 95% of the number of views in the second week. If we knew the number of views in the second week, we could calculate the number of views in the third week by multiplying by 0.95.

However, since we don't know the number of views in the second week, we can't calculate the exact number of views in the third week. We can only say that the number of views in the third week was 5% less than the number of views in the second week.

Therefore, we can write the following equation:

0.95x = (number of views in the third week)

This equation tells us that if we know the number of views in the second week (x), we can calculate the number of views in the third week by multiplying by 0.95.

However, since we don't know the number of views in the second week, we can't calculate the exact number of views in the third week.

To know more about equation visit:

https://brainly.com/question/29538993

#SPJ11

ANSWER:

D. 44

STEP-BY-STEP EXPLANATION:

The percentage of a number is calculated by multiplying the percentage number in its decimal form by the number whose percentage is to be calculated.

A percentage number is converted to a decimal by dividing that number by 100, so it would finally look like this:

So the correct answer is D. 44

Answer:

From the said lesson, the difficulty that I have been trough in dealing over the exponential expressions is the confusion that frequently occurs across my system whenever there's a thing that I haven't fully understand. It's not that I did not actually understand what the topic was, but it is just somewhat confusing and such. Also, upon working with exponential expressions — indeed, I have to remember the rules that pertain to dealing with exponents and frequently, I will just found myself unconsciously forgetting what those rule were — rules which is a big deal or a big thing in the said lesson because it is obviously necessary/needed over that matter. Surely, it is also a big help for me to deal with exponential expressions since it's so much necessary — it's so much necessary but I keep fogetting it.. hence, that's why I call it a difficulty. That's what my difficulty. And in order to overcome that difficulty, I will do my best to remember and understand well the said rules as soon as possible.

Christopher's statement "whenever you multiply a whole number by a fraction the product is either greater or less than the whole number" is correct.

A fraction is a way to describe a part of a whole. such as the fraction ¼ can be described as 0.25.

The statement made by Christopher is correct, this is because a fraction is always either greater or lesser than 1. And multiplying a fraction to a whole number will give result either greater or less than the whole number, depending upon the type of fraction it is.

For example let's take the number 100,

Let's take a proper fraction, 2/5. Therefore, the result will be,

100 × (2/5) = 40

Let's take an improper fraction, 7/5. Therefore, the result will be,

100 × (7/5) = 140

Hence, Christopher's statement is correct.

Learn more about Fraction here:

https://brainly.com/question/1301963

#SPJ1

9514 1404 393

Answer:

  no solution (parallel lines)

Step-by-step explanation:

You observe that the coefficient in the second equation are all multiples of 3. Dividing that equation by -3 gives the two equations ...

The slopes are the same (2) and the intercepts are different (-11, 5), so the lines are parallel. There is no solution.

Answer:

Step-by-step explanation:

To solve the given equation \(\sf x - y = 4 \\\), we can perform the following calculations:

a) To find the value of \(\sf 3(x - y) \\\):

\(\sf 3(x - y) = 3 \cdot 4 = 12 \\\)

b) To find the value of \(\sf 6x - 6y \\\):

\(\sf 6x - 6y = 6(x - y) = 6 \cdot 4 = 24 \\\)

c) To find the value of \(\sf y - x \\\):

\(\sf y - x = - (x - y) = -4 \\\)

Therefore:

a) The value of \(\sf 3(x - y) \\\) is 12.

b) The value of \(\sf 6x - 6y \\\) is 24.

c) The value of \(\sf y - x \\\) is -4.

\(\huge{\mathfrak{\colorbox{black}{\textcolor{lime}{I\:hope\:this\:helps\:!\:\:}}}}\)

♥️ \(\large{\underline{\textcolor{red}{\mathcal{SUMIT\:\:ROY\:\:(:\:\:}}}}\)

this expression, we'll use the property log(a/b) = log(a) - log(b):
log6(x/5) = log6(x) - log6(5)

(2) log2(x(y½))

For this expression, we'll use two properties: log(ab) = log(a) + log(b) and log(a^b) = b*log(a):
log2(x(y½)) = log2(x) + log2(y½)
Now apply the second property:
log2(x) + (1/2)*log2(y)

(b) Use the properties of logarithms to rewrite and simplify the logarithmic expression.
log3(92 · 24)

First, we'll use the property log(ab) = log(a) + log(b):
log3(92 · 24) = log3(92) + log3(24)

(c) Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.)
log4(xy⁴z⁴)

We'll use the properties log(ab) = log(a) + log(b) and log(a^b) = b*log(a):
log4(xy⁴z⁴) = log4(x) + log4(y⁴) + log4(z⁴)
Now apply the second property:
log4(x) + 4*log4(y) + 4*log4(z)

To know more about lograthim visit

https://brainly.com/question/25710806

#SPJ11

Answer:

n

Step-by-step explanation:

Answer:

\(\fbox{\begin{minipage}{11em}N is closer to P than M\end{minipage}}\)

Step-by-step explanation:

Step 1: Define the way to calculate distance between 2 points in two-dimensional (2D) plane

Supposing that there are two points \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\) on 2D plane.

The distance \(d\) between these two points is calculated by:

\(d = \sqrt{(x_{1} - x_{2}) ^{2} + (y_{1} - y_{2})^{2} }\)

Step 2: Calculate the distance \(d_{1}\) between \(M(3, 6)\) & \(P(-2, -1)\) and distance \(d_{2}\) between \(N(6, -4)\) and \(P(-2, -1)\)

Applying the formula in step 1:

\(d_{1} = \sqrt{(3+ 2) ^{2} + (6 + 1)^{2} } = \sqrt{25 + 49} = \sqrt{74}\)

\(d_{2} = \sqrt{(6+ 2) ^{2} + (-4 + 1)^{2} } = \sqrt{64 + 9} = \sqrt{73}\)

Step 3: Compare and conclude

Because \(\sqrt{73} < \sqrt{74}\)  =>  \(d_{2} < d_{1}\) => N is closer to P than M

Hope this helps!

:)

Answer:

You Answer is C. 738

Step-by-step explanation:

L x W x H

A= 2 w  l   h  l  h  w  = 2  18 13.5 4  13.5 4 18  =  738

The solution to the given recurrence relation is (c): ak = (3^k + 1) + 2^k + 1

The solution of the given recurrence relation can be found by substitution. We can substitute the given formula for ak into the equation for a_(k+1) and see if the two are equal. If they are equal, then we have the solution.

Starting with the equation for ak:

a_k = 2a_(k-1) + 3^k

Substituting the formula for ak-1:

a_k = 2(2a_(k-2) + 3^{k-1}) + 3^k

Expanding the right side:

a_k = 4 a_(k-2) + 2 * 3^{k-1} + 3^k

Continuing this process, we can see that the solution is:

a_k = 2^(k) a₀ + (2^{k-1} + 2^{k-2} + ... + 2^1) * 3^{k-1} + 3^k

Since the series 2^k-1 + 2^{k-1} + ... + 2^1 = 2^k - 1, we have:

a_k = 2^(k) a₀ + (2^k - 1) * 3^{k-1} + 3^k

Substituting the initial value a₀ = 5:

a_k = 2^(k) * 5 + (2^k - 1) * 3^{k-1} + 3^k

Simplifying:

a_k = 2^{k} (5 + 3^{k -1}) - 3^{k-1}(1 + 3)

a_k =  2^{k} (5 + 3^{k -1}) - 4* 3^{k-1}

a_k = 5* 2^{k} + 3^{k -1}( 2^{k}  - 4)

So, the solution to the given recurrence relation is (c):

ak = (3^k + 1) + 2^k + 1

To know more on recurrence relation

https://brainly.com/question/9521757

#SPJ4

The correct answer is B

Answer: always

Step-by-step explanation:

Answer:

Hey there!

f(x)=5x+3

f(5)=5(5)+3

f(5)=25+3

f(5)=28

Let me know if this helps :)

Answer:

f(5) =28

Step-by-step explanation:

\(f(x) = 5x + 3\\f(5) =?\\\\\)

Substitute 5 for x in the given function

\(f(5) = 5(5) +3\\f(5) = 25+3\\f(5) =28\)

Answer:

25.12  pulgadas

Step-by-step explanation:

Answer: It would be (-11/2,9/2)

Step-by-step explanation: Use the midpoint formula to calculate the answer, it is (x1+x2)/2,(y1+y2)/2

Answer:

(-5.5, 4.5)

Step-by-step explanation:

(x2+x1)/2

(y2+y1)/2

-8+-3=-11/2=-5.5

6+3=9/2=4.5

The formula that can be used to describe the sequence is:\(a(n) = (-1)^(n+1) * 3^(n) * 4.\)

The given sequence is -81, 108, -144, 192.

The formula that can be used to describe the sequence is:  \(a(n) = (-1)^(n+1) * 3^(n) * 4\), where n is the nth term in the sequence.

This formula is a geometric sequence formula that can be used to describe the given sequence.

The formula represents the nth term of the sequence as a function of the position of the term in the sequence.

Here, n represents the position of the term in the sequence

.For the given sequence, the first term is -81, which corresponds to the first position in the sequence (n = 1).

The second term is 108, which corresponds to the second position in the sequence (n = 2).

The third term is -144, which corresponds to the third position in the sequence (n = 3).

The fourth term is 192, which corresponds to the fourth position in the sequence (n = 4).

Therefore, the formula that can be used to describe the sequence is: \(a(n) = (-1)^(n+1) * 3^(n) * 4.\)

Know more about sequence  here:

https://brainly.com/question/7882626

#SPJ11

Answer:

B. 1.00 x 10 to the 4th power

Step-by-step explanation:

The instructor can conclude that the class performed around the average and that the distribution of grades is likely to be symmetrical. This can also imply that there is no skewness or bias in the data. Answer more than 100 words:When an instructor notes that the mean, median, and mode are all exactly 76.00, it can be inferred that the distribution of grades in the class is roughly symmetrical. This indicates that there are nearly equal numbers of students above and below the average grade.

The mean, median, and mode are three measures of central tendency that are used to summarize the distribution of a dataset. The mean is calculated by adding all the values in the dataset and dividing by the number of observations, while the median is the middle value when the dataset is ordered from smallest to largest. The mode is the value that occurs most frequently in the dataset. When all three measures of central tendency are equal, as in this case, it means that the dataset is roughly symmetrical and there is no skewness or bias. This can also indicate that the dataset is normally distributed, which means that it follows a bell-shaped curve.The instructor can conclude that the class performed around the average, as the mean, median, and mode are all 76.00.

However, it is important to note that this does not give any information about the spread of the dataset. In other words, the instructor cannot tell from this information alone whether the grades were tightly clustered around the average or widely dispersed. For this reason, it may be useful to calculate additional measures of spread, such as the range or standard deviation, to get a better understanding of the distribution.

The instructor can conclude that the class performed around the average and that the distribution of grades is likely to be symmetrical, with no skewness or bias. However, it is important to remember that this information alone does not provide any insight into the spread of the dataset. Additional measures of spread would need to be calculated to determine the degree of variability in the grades.

To know more about mean, median, and mode visit:

brainly.com/question/21035124

#SPJ11

Natural selection will happen if three factors are met according to Darwin's theory. These circumstances, which are shown in bold above, compete with one another for survival, variation, and inheritance.

Learn more about Natural selection refer to :

https://brainly.com/question/23929271

#SPJ4

Answer:

20 weeks

Step-by-step explanation:

The calculation of The number of weeks it will take to save £360 is given below:

Now  $250 should be divided into a ratio of 3:2

He keeps = 150

And, he gives her = 100

As he saves 12% of his share

i.e.

= 12% of 150

= 18

Now the number of weeks is

= 360 ÷ 18

= 20 weeks

The probability questions related to the weights of dogs in a dog show can be solved using the standard normal distribution curve.

The weights of dogs in a dog show follow a normal distribution with a mean of 58 pounds and a standard deviation of 17.2 pounds. By using the standard normal distribution, we can standardize the values and calculate probabilities using z-scores.

To find probabilities, we need to convert the given values into z-scores using the formula:

z = (x - μ) / σ

Where:

z is the z-score

x is the given value

μ is the mean

σ is the standard deviation

Once we have the z-score, we can use a standard normal distribution table or a calculator to find the corresponding probabilities.

For example, if we want to find the probability of a dog weighing less than 65 pounds, we calculate the z-score as:

z = (65 - 58) / 17.2 ≈ 0.407

Using the standard normal distribution table or calculator, we can find the probability corresponding to a z-score of 0.407, which represents the area to the left of the z-score. Similarly, we can calculate probabilities for other scenarios such as finding the probability of a dog weighing between two specific weights or exceeding a certain weight.

Learn more about standard normal distribution here: brainly.com/question/30390016

#SPJ11

Given a linear system of equations in unknowns x, y, and z with the following augmented matrix:{[1, -1, 0, 0, -7], [-2, 3, 0, 0, 2], [0, 0, 4, -2, 2]}Use Gauss-Jordan elimination to solve the system for x, y, and z.Solution:Step 1. The first step in solving this linear system of equations is to write the matrix in the form of an augmented matrix. In the following, we list the system of equations associated with the augmented matrix: 1x−1y=−72x+3y=24z−y=1  We begin by focusing on the first equation, which is:1x−1y=−7.

To get rid of the x-coefficient, we add one time the first equation to the second equation. This operation is written as follows:{[1, -1, 0, 0, -7], [-2, 3, 0, 0, 2], [0, 0, 4, -2, 2]}We add row1 to row2. -2r1 + r2 = r2{-2, 2, 0, 0, 14}r3 = r3This gives us the new augmented matrix.{[1, -1, 0, 0, -7], [0, 1, 0, 0, -5], [0, 0, 4, -2, 2]}Step 2Next, we focus on the second equation:0x+1y=−5.

The y-variable is isolated, and we now look at the third equation.4z−2y=1We can isolate the variable z by dividing the entire equation by 4 as follows:z−0.5y=0.25In order to eliminate y in the third row, we add 0.5 times the second row to the third row. This operation is written as follows:{[1, -1, 0, 0, -7], [0, 1, 0, 0, -5], [0, 0, 4, -2, 2]}We add 0.5 r2 to r3. r3 + 0.5r2 = r3{[1, -1, 0, 0, -7], [0, 1, 0, 0, -5], [0, 0, 4, -1, -1]}Step 3We can now solve for z using the third equation:4z−1y=−1z = (-1 + y) / 4.

Substituting this into the second equation gives:-2((1 - y) / 4) + 3y = 2y - 1 = 2y - 1Thus, y = 1/2.Substituting the value of y = 1/2 into the first equation gives:x - (1/2) = -7, so x = -13/2.Finally, we can substitute the values of x and y into the third equation to get the value of z: 4z - 1(1/2) = -1, so z = -3/2.The solution to the system of linear equations is: x = -13/2, y = 1/2, and z = -3/2.

Learn more about linear system of equations:

https://brainly.com/question/13729904

#SPJ11

Answer:

The Median is 75

Step-by-step explanation:

median is the middle number in a set of given numbers

arranging in order will be

30,64,70,80,90,110

the middle numbers are 70 and 80

Median=80+70/2

=150/2

Median=75

The probability that an irs auditor will catch only 2 income tax returns with illegitimate deduc-tions = 0.24

Let us assume that X be the number of returns containing illegitimate deductions in the sample.

Here, X has a hypergeometric distribution.

We need to find the probability that an IRS auditor will catch only 2 income tax returns with illegitimate deduc-tions.

Here, N = 15, r = 9, n = 5

So, P (X = 2) = (⁹C₂ × ⁶C₃) / (¹⁵C₅)

We know that the combination formula:

⁹C₂ = 9! / (2! × 7!)

     = 36

⁶C₃ = 6! / (3! × 3!)

     = 20

¹⁵C₅ = 15! / (5! 10!)

      = 3003

P (X = 2) = (⁹C₂ × ⁶C₃) / (¹⁵C₅)

              = (36 × 20) / (3003)

              = 0.24

Therefore, the required probability is: 0.24

Learn more about the probability here:

https://brainly.com/question/15124899

#SPJ4

Answer:

hhbnnnnkkkkkkkkkkkkkijkkiikkkk