Suppose z=x^2*sin(y), x=2s^2−4t^2, y=4st.
A. Use the chain rule to find ∂z/∂s and ∂z/∂t as functions of x, y, s and t.

To find variable values and angle measures, identify knowns, use appropriate equations for unknowns, calculate vector components, use trigonometric functions and the Pythagorean theorem for the resultant vector, and ensure angles are in radians.

To find the values of the variables and the measures of the angles, you need to follow a systematic approach. Start by making a list of known variables and inferences from the problem. Once you've identified what is given, you can solve for the unknowns using the appropriate equations. If the problem involves vectors, you would identify the x- and y-axes before finding the vector components Ax = A cos and Ay = A sin, where A is the magnitude of the vector and cos and sin are the trigonometric functions corresponding to the angles the vectors make with the x-axis. After computing these, you can use the Pythagorean theorem to find the resultant vector length.

To verify your solutions, including the measures of the angles in radians, it's important to assess whether the answers are reasonable. This may involve checking the magnitude and direction of the resultant vector and ensuring that the angles are within the possible range (0 to 2π radians for a full circle).

Answer:

The correct option is:

C. 5,418.54 feet above sea level

Step-by-step explanation:

Nadia started at 19.26 feet below sea level.

She then changed her altitude by climbing a total of 5,437.8 feet up from her initial position.

Her new position would be 5437.8-19.26 feet above the sea level

i.e.  5,418.54 feet above sea level

Hence, the correct option is:

C. 5,418.54 feet above sea level

Answer:

5,418.54

Step-by-step explanation:

- Independent quantity: Number of hours worked per week

- Dependent quantity: Earnings per week

- Domain: 20 ≤ hours ≤ 25

- Range: $180 ≤ earnings ≤ $225

In this situation:

- Independent quantity: The number of hours Wendy plans to work per week

- Dependent quantity: The amount of money she earns per week

Reasonable domain and range values:

- Domain: The reasonable range of values for the number of hours Wendy works per week is between 20 and 25 hours. This is because she plans to work between 20 and 25 hours per week.

- Range: The amount of money Wendy earns per week depends on the number of hours she works. The range of values for her earnings can be calculated by multiplying the number of hours worked by her hourly rate of $9. Therefore, the range of values for her earnings per week will be between $180 (20 hours * $9) and $225 (25 hours * $9).

To summarize:

- Independent quantity: Number of hours worked per week

- Dependent quantity: Earnings per week

- Domain: 20 ≤ hours ≤ 25

- Range: $180 ≤ earnings ≤ $225

Answer:

563

Step-by-step explanation:

Plato

Answer:

8.99

Step-by-step explanation:

The ordered pairs (0,0), (1,4), (2,16), (3,36), and (4,64) can be represented by the function y = x^2.

The question involves a list of ordered pairs that represent a mathematical function. Here, the ordered pairs are (0,0), (1,4), (2,16), (3,36), (4,64). Through these pairs, we can see a relationship between the x-values (first number in each ordered pair) and their respective y-values (second number in each pair). Specifically, each y-value appears to be the square of its respective x-value. By noticing this pattern, we can write a rule that represents this function: y = x^2. This means that for any given x-value, you just need to square it to get its corresponding y-value.

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The perimeter of the figure is 58 feet if the missing side length is 10 feet.

It is defined as the length of the circumference or outline of the geometry. It is the sum of the all side measures.

We have a two-dimensional geometry in the picture with dimensions.

The missing side length  = 15 - 5 = 10 ft

The perimeter of the figure = 5+7+10+10+15+11

= 58 feet

Thus, the perimeter of the figure is 58 feet if the missing side length is 10 feet.

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Answer: Graph D will be correct graph for the given function.

Explanation:

Given function [tex]f(x) = x^4-2x^3-3x^2+4x+1[/tex]

Since it is a bi-quadratic equation thus it must have 4 roots and (0,1) is one of its point.

Moreover, the degree of the function is even thus the end behavior of the function is[tex]f(x)\to+\infty[/tex], as [tex]x\to-\infty[/tex] and [tex]f(x)\to+\infty[/tex] as [tex]x\to+\infty[/tex]

In graph A, function has four root but it does not have the end behavior same as function f(x).( because in this graph [tex]f(x)\to-\infty[/tex], as [tex]x\to-\infty[/tex] and [tex]f(x)\to-\infty[/tex], as [tex]x\to+\infty[/tex].) so, it can not be the graph of given function.

In graph B,  neither  it has four root nor it has the end behavior same as function f(x).(because in this graph [tex]f(x)\to+\infty[/tex] as  [tex]x\to-\infty[/tex] and [tex]f(x)\to-\infty[/tex] as [tex]x\to+\infty[/tex].) so, it can not be the graph of given function.

In graph C, neither  it has four root nor it has the same end behavior as function f(x).(because in this graph [tex]f(x)\to-\infty[/tex] as  [tex]x\to-\infty[/tex] and [tex]f(x)\to+\infty[/tex] as[tex]x\to\infty[/tex].) so, it also can not be the graph of given function.

In graph D it has four root as well as it has the same end behavior as the given function. Also it passes through the point (0,1).

Thus, graph D is the graph of given function.

Answer:

Luis, because his sample is taken from the population of all seventh graders. However, Samanthan has the advantage of having a bigger sample, and that makes it more representative.

Step-by-step explanation:

Taking the sample of the library as Samantha did, is biased, because most of the students there is checking a book, and the aim of the study is not only the ones that use the library, but in general the seventh grade students.

I guess there should be an options to choose. But I'm pretty sure that the answer is: To set up long division of polynomials, you should make sure that each polynomial is written in descending order and has no missing terms.

Step [tex]1[/tex]

Convert mixed numbers to improper fractions

we know that

[tex]4\frac{1}{4}=\frac{4*4+1}{4}=\frac{17}{4}[/tex]

[tex]2\frac{3}{4}=\frac{2*4+3}{4}=\frac{11}{4}[/tex]

Step [tex]2[/tex]

Find the difference of the numbers

[tex]\frac{17}{4}-\frac{11}{4}=\frac{6}{4}=\frac{3}{2}[/tex]

Step [tex]3[/tex]

Convert improper fraction to mixed number

[tex]\frac{3}{2}=1.5=1+0.5=1+\frac{1}{2} =1\frac{1}{2}[/tex]

therefore

the answer is the option B

[tex]1\frac{1}{2}[/tex]

The question's topic is about converting volume from cups to quarts. In this particular case, 15 cups of chicken stock is equal to 3.75 quarts.

The subject of this question is the conversion of volume from cups to quarts. In this case, the chicken soup recipe calling for 15 cups of chicken stock needs to be converted into quarts. We know that 1 quart is equal to 4 cups, so we can set up the equation:

4 cups/1 quart = 15 cups/x quarts

To solve for x, we multiply 15 (cups) by 1 (quart) and then divide by 4 (the number of cups in a quart).

Therefore, x = (15*1)/4 = 3.75 quarts. So, 15 cups of chicken stock is equal to 3.75 quarts.

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The height of the sail in feet is 24 feet.

According to the Pythagoras theorem, the square of the hypotenuse of a triangle, which has a straight angle of 90 degrees, equals the sum of the squares of the other two sides.

Look at the triangle ABC below, where we have

BC² = AB² + AC²​​.

The base is represented by AB, the altitude by AC, and the hypotenuse by BC in this equation. It should be remembered that a right-angled triangle's hypotenuse is its longest side.

Given:

Base = 10 feet

Hypotenuse = 26 feet

Using Pythagoras theorem

H² = P² + B²

P²= (24)² - (10)²

P² = 676 - 100

P² = 576

P=√576

P= 24 feet

hence, height of the sail in feet is 24 feet.

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The unit rate is calculated by dividing the distance run by the time taken. In this case, the unit rate is 5.4 miles per hour, indicating that the student runs 5.4 miles for every hour they run.

To find the unit rate, we first convert the distance to a decimal. So, 3 3/5 miles becomes 3.6 miles. Unit rate is the ratio of the amount of a single unit of an item to its cost. In this case, your unit rate is the distance you run (in miles) per unit of time (in minutes).

Now, you should divide the total distance (3.6 miles) by the time (40 minutes). 3.6 miles / 40 minutes = 0.09 miles per minute.

If you want the unit rate per hour (commonly used), you should multiply this result by 60 (as there are 60 minutes in an hour). 0.09 miles per minute * 60 minutes/hour = 5.4 miles/hour.

So, your unit rate is 5.4 miles per hour, meaning you run 5.4 miles for every one hour of running.

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To find out the different orders four cars can finish a race, we calculate the permutations, which is 4 factorial, resulting in 24 different orders.

The student is asking about the number of different ways that four cars can finish a pinewood derby race, which is a question of permutations in mathematics. In this case, we want to find out the number of possible arrangements for the four cars crossing the finish line. Since each car is distinct, the number of permutations of four items is given by 4! (4 factorial), which is calculated as 4 × 3 × 2 × 1.

Therefore, there are 24 different orders in which the cars can cross the finish line. Remember, factorial is a mathematical operation that involves multiplying a sequence of descending natural numbers.

The area and the circumference of the given circle are 38.5 cm² and 20 cm respectively.

A circle is a round-shaped figure that has no corners or edges. In geometry, a circle can be defined as a closed, two-dimensional curved shape.

Given that, a circle with radius 3.5 cm, we are asked to find the circumference and the area of the given circle,

Area of a circle = π × radius²

Circumference of a circle = 2π × radius

1) Area =

π × 3.5²

= 3.14 × 12.25

= 38.465

≈ 38.5 cm²

2) Circumference =

2π × 3.5

= 2 × 3.14 × 3.5

= 21.98

≈ 20 cm

Hence, the area and the circumference of the given circle are 38.5 cm² and 20 cm respectively.

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