Shawn wrote down the activities for his day on Saturday. In which situation will his activity result in a final value of zero?
1 point
A. Shawn places four quarters in a jar of quarters which contains four quarters.
B. In the morning, Shawn added six hard candies to a jar which contained four hard candies. By the end of the day he ate ten candies from this jar.
C. Shawn starts out on the ground and then climbs ten feet on a ladder.
D. Shawn travels east ten feet and then travels south ten feet

Answers

Answer 1

The situation in which Shawn's activity will result in a final value of zero is Shawn travels east ten feet and then travels south ten feet. The correct option is D.

This is because when Shawn travels east ten feet, he moves horizontally to the right of his starting point. When he travels south ten feet after that, he moves vertically downwards from his previous position, cancelling out the horizontal movement he made earlier.

The displacement caused by Shawn's movement in the east direction is equal in magnitude but opposite in direction to the displacement caused by his movement in the south direction.

The net displacement of Shawn's movement is zero, and he ends up back at his starting point. Options A, B, and C do not involve any movements that result in a net displacement of zero.

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Related Questions

Julie guessed at random for each question on a true false quiz of ten questions. What is the

probability that she got exactly seven questions correct?

3078

Answers

The probability that Julie got exactly seven questions correct is approximately 0.117 or 11.7%.

The probability of guessing the correct answer on any single true-false question is 1/2, and the probability of guessing the wrong answer is also 1/2.

The probability of guessing exactly seven questions correctly can be calculated using the binomial distribution formula:

P(X = k) = (n choose k) * p^k * (1-p)^(n-k)

where:

n is the total number of questions, which is 10 in this case.

k is the number of questions guessed correctly, which is 7 in this case.

p is the probability of guessing any individual question correctly, which is 1/2 in this case.

Using the formula, we get:

P(X = 7) = (10 choose 7) * (1/2)^7 * (1/2)^(10-7)

= 120 * (1/2)^10

= 0.1171875

Therefore, the probability that Julie got exactly seven questions correct is approximately 0.117 or 11.7%.

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A portion of a truss vehicle bridge has steel
beams that form an isosceles triangle with the
dimensions shown. A pedestrian handrail is
attached to the side of the bridge so that it is
parallel to the road.
A
24 feet
28.5 feet
30 feet
What is the height, x, of the handrail above th
road to the nearest tenth of a foot?

Answers

Answer:

3.5 feet

Step-by-step explanation:

You want the difference in height between similar isosceles triangles, one with height of 30 ft and a base of 24 ft, the other with a side length of 28.5 ft.

Relations

We can find the side length of the larger triangle using the Pythagorean theorem. It will be ...

longer side = √(30² +12²) ≈ 32.311 ft

Similar triangles

Then the length x is the difference between the altitudes of the triangles. The altitudes are proportional to the side lengths, so we have ...

(30 -x)/28.5 = 30/32.311

x = 30-(28.5)(30/32.311) = 30(1 -28.5/32.311) ≈ 3.538 ≈ 3.5 . . . . feet

The hand rail is about 3.5 feet above the bridge deck.

Trigonometry

We recognize that the distance from the hand rail to the top of the triangle is the product of the given side length (28.5 ft) and the cosine of the angle between the side and the altitude.

The tangent of that angle is the ratio of its opposite side (12 ft) to its adjacent side (30 ft), or θ = arctan(12/30).

The value of x is the difference of the altitudes of the triangles, so is ...

x = 30 -28.5·cos(arctan(12/30)) ≈ 3.5 ft

We find this easier to compute.

#95141404393

Grady is comparing three investment accounts offering different rates.

Account A: APR of 4. 95% compounding monthly

Account B: APR of 4. 85% compounding quarterly

Account C: APR of 4. 75% compounding daily Which account will give Grady at least a 5% annual yield? (4 points)

Group of answer choices

Account A

Account B

Account C

Account B and Account C

Answers

The account that will give Grady at least a 5% annual yield is Account C

Why account C will give Grady at least a 5% annual yield?

We can use the formula for compound interest to compare the three investment accounts and find the one that will give Grady at least a 5% annual yield:

FV = PV × (1 + r/n)^(n*t)

where FV is the future value, PV is the present value, r is the annual interest rate as a decimal, n is the number of times the interest is compounded per year, and t is the number of years.

For Account A:

APR = 4.95%, compounded monthly

r = 0.0495

n = 12

t = 1

FV = PV × (1 + r/n)^(nt)

FV = PV × (1 + 0.0495/12)^(121)

FV = PV × 1.050452

To get at least a 5% annual yield, we need FV/PV ≥ 1.05

1.050452/PV ≥ 1.05

PV ≤ 1.000497

Therefore, Account A will not give Grady at least a 5% annual yield.

For Account B:

APR = 4.85%, compounded quarterly

r = 0.0485

n = 4

t = 1

FV = PV × (1 + r/n)^(nt)

FV = PV × (1 + 0.0485/4)^(41)

FV = PV × 1.049375

To get at least a 5% annual yield, we need FV/PV ≥ 1.05

1.049375/PV ≥ 1.05

PV ≤ 1.000351

Therefore, Account B will not give Grady at least a 5% annual yield.

For Account C:

APR = 4.75%, compounded daily

r = 0.0475

n = 365

t = 1

FV = PV × (1 + r/n)^(nt)

FV = PV × (1 + 0.0475/365)^(3651)

FV = PV × 1.049038

To get at least a 5% annual yield, we need FV/PV ≥ 1.05

1.049038/PV ≥ 1.05

PV ≤ 1.000525

Therefore, Account C will give Grady at least a 5% annual yield.

Therefore, the account that will give Grady at least a 5% annual yield is Account C.

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PLEASE HELP WITH 4 AND 5

Answers

1. The area of the shaded region is

2. percentage of the shaded part is 89.6%

What is area of shape?

The area of a shape is the space occupied by the boundary of a plane figures like circles, rectangles, and triangles.

The area of the shaded part = area of the rectangle - area of unshaded part

Area of rectangle = 7× 11 = 77 unit²

area of rectangle = 1/2 bh

= 1/2 × 4 × 4

= 1/2 × 8

= 4 unit²

area of second triangle = 4 units²

area of unshaded part = 4+4 = 8 units²

area of shaded part = 77-8 = 69units²

2. percentage of the rectangle shaded = 69/77 × 100

= 6900/77 = 89.6%

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Suppose that a report by a leading medical organization claims that the healthy human heart beats an average of 72 times per minute. Advances in science have led some researchers to question if the healthy human heart beats an entirely different amount of time, on average, per minute. They obtain pulse rate data from a sample of 85 healthy adults and find the average number of heart beats per minute to be 76, with a standard deviation of 13. Before conducting a statistical test of significance, this outcome needs to be converted to a standard score, or a test statistic. What would that test statistic be

Answers

The test statistic (or standard score) would be 1.72.

To convert the outcome of the pulse rate data to a standard score, we would need to calculate the z-score. The formula for the z-score is: (sample mean - population mean) / (standard deviation / square root of sample size).

In this case, the sample mean is 76, the population mean (according to the report) is 72, the standard deviation is 13, and the sample size is 85. Plugging these values into the formula, we get:

(76 - 72) / (13 / sqrt(85)) = 1.72.

Therefore, the test statistic (or standard score) is 1.72. This indicates that the sample mean of 76 is 1.72 standard deviations above the population mean of 72. This information can be used to conduct a statistical test of significance and determine whether the difference between the sample mean and population mean is statistically significant.

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Classify triangle ABD by its sides and then by its angles.

Select the correct terms from the drop-down menus.

Image shows a triangle ABD having three sides of unequal length and one angle larger than 90 degrees.

Triangle ABD is

and

.
.

Answers

Triangle ABD is scalene and obtuse.

How to classify the triangle?

The triangle has three sides of unequal length, hence it is classified as an scalene triangle.

(it would be equilateral if all had the same length, and isosceles if two sides have the same length).

The triangle has one angle larger than 90º, hence it is classified as an obtuse triangle.

(acute with no angles of 90º or greater, right with one angle of exactly 90º).

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What is the maximum height of Anna’s golf ball? The equation is y=x-0. 04x^2.

The maximum height is____ feet

Answers

The maximum height of Anna's golf ball is 6.25 feet.

To find the maximum height of Anna's golf ball, we need to determine the vertex of the parabolic equation y = x - 0.04x^2. The x-coordinate of the vertex can be found using the formula:

x = -b / (2a)

In this case, the coefficients a and b are:
a = -0.04
b = 1

Substituting the values into the formula:

x = -1 / (2 * -0.04)
x = -1 / (-0.08)
x = 12.5

Now, we need to find the y-coordinate of the vertex by plugging the x-coordinate back into the equation:

y = 12.5 - 0.04(12.5)^2

y = 12.5 - 0.04(156.25)

y = 12.5 - 6.25

y = 6.25

So, the maximum height of Anna's golf ball is 6.25 feet.

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2. Assume that a cell is a sphere with radius 10 or 0. 001 centimeter, and that a cell's density is 1. 1 grams per cubic centimeter. A. Koalas weigh 6 kilograms on average. How many cells are in the average koala?

Answers

The number of cells found in an average Koala is 1.30 x 10¹², under the condition that a cell is a sphere with radius 10 or 0. 001 centimeter.

Then the volume of a sphere with radius 10 cm is considered to be 4/3π(10)³ cubic cm that is approximately 4,188.79 cubic cm.
The evaluated volume of a sphere with radius 0.001 cm is 4/3π(0.001)³ cubic cm that is approximately 0.00000419 cubic cm.

Then the evaluated mass of a single cell is found by applying the formula
mass = density x volume
In case of larger cell, the mass will be
mass = 1.1 g/cm³ x 4,188.79 cubic cm
= 4,607.67 grams

In case of smaller cell, the mass will be

mass = 1.1 g/cm³ x 0.00000419 cubic cm
= 0.00000461 grams

As koalas measure an average of 6 kilograms or 6,000 grams², we can evaluate the number of cells in an average koala using division of the weight of the koala by the mass of a single cell

In case of larger cells
number of cells = weight of koala / mass of single cell
number of cells = 6,000 grams / 4,607.67 grams
≈ 1.30 x 10⁶ cells

For smaller cells:
number of cells = weight of koala / mass of single cell
number of cells = 6,000 grams / 0.00000461 grams

≈ 1.30 x 10¹² cells
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⁶

Q1. Consider the following options for characters in setting a password:
.
.

Digits = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

Letters = { a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, V, W, X, Y, z}

Special characters = 1 *, &, $. #}

Compute the number of passwords possible that satisfy these conditions:

• Password must be of length 6.

Characters can be special characters, digits, or letters,

Characters may be repeated.
.

Answers

There are 4,096,000,000 possible passwords of length 6 using special characters, digits, and letters, with characters allowed to be repeated.

To compute the number of passwords possible with a length of 6 using digits, letters, and special characters, with characters allowed to be repeated, follow these steps:

1. Count the number of options for each character type:
- Digits: 10 (0-9)
- Letters: 26 (a-z)
- Special characters: 4 (*, &, $, #)

2. Combine the options for all character types:
Total options per character = 10 digits + 26 letters + 4 special characters = 40

3. Calculate the number of possible passwords:
Since characters may be repeated and the password has a length of 6, the number of possible passwords = 40^6 (40 options for each of the 6 character positions)

4. Calculate the result:
Number of possible passwords = 40^6 = 4,096,000,000

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A number cube is rolled twice. what is the probability of getting a six on the first role, then a number less than 5 on the second roll?

Answers

If a number cube is rolled twice, then the probability of getting a six on the first role and then a number less than 5 on the second role is 1/9.

To find the probability of getting a six on the first roll and a number less than 5 on the second roll, we will multiply the individual probabilities of each event.

A number cube has 6 faces, so the probability of rolling a six is 1/6.

For the second roll, there are 4 numbers less than 5 (1, 2, 3, and 4), so the probability of rolling a number less than 5 is 4/6 or 2/3.

To find the combined probability, simply multiply the two probabilities: (1/6) × (2/3) = 2/18 = 1/9.

So, the probability of getting a six on the first roll and a number less than 5 on the second roll is 1/9.

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Find the volume of a pyramid with a square base, where the side length of the base is

11. 8 ft and the height of the pyramid is 5. 2 ft. Round your answer to the nearest

tenth of a cubic foot.

Answers

The volume of the pyramid with a square base of side length 11.8 ft and a height of 5.2 ft is 240.0 cubic feet.

To find the volume of a pyramid with a square base of side length 11.8 ft and a height of 5.2 ft, you can use the following formula:

Volume = (1/3) × Base Area × Height

1: Find the base area.

The base is a square with a side length of 11.8 ft, so the area of the base is:

Base Area = Side Length × Side Length

Base Area = 11.8 ft × 11.8 ft

Base Area ≈ 139.24 square ft

2: Find the volume.

Now, use the formula to find the volume:

Volume = (1/3) × Base Area × Height

Volume = (1/3) × 139.24 sq ft × 5.2 ft

Volume ≈ 240.0368 cubic ft

3: Round your answer to the nearest tenth.

Volume ≈ 240.0 cubic ft

So, the volume of the pyramid is approximately 240.0 cubic feet.

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Trevor is comparing two mortgage options from two different banks for his 20 year $120,000 mortgage. He thinks both mortgages are pretty much the same and is having a hard time deciding which bank to partner with. Bank A: 5% with monthly payments of $791. 95 Bank B: 4. 75% with monthly payments of $775. 47

Answers

Bank B is offering a lower interest rate and will result in a lower total cost over the 20-year period. Even though the monthly payment is slightly lower with Bank B, Trevor should choose Bank B because he will save money in the long run due to the lower interest rate.

Which bank should Taravar choose? in which he will save money in the long run due to the lower interest rate.

To compare the two mortgage options, Trevor needs to consider both the interest rate and the monthly payment amount.

Bank A offers a 5% interest rate with a monthly payment of $791.95. The total amount he will pay over 20 years is:

$791.95 x 12 months/year x 20 years = $190,068

Bank B offers a 4.75% interest rate with a monthly payment of $775.47. The total amount he will pay over 20 years is:

$775.47 x 12 months/year x 20 years = $186,113.60

So, in this case, Bank B is offering a lower interest rate and will result in a lower total cost over the 20-year period. Even though the monthly payment is slightly lower with Bank B, Trevor should choose Bank B because he will save money in the long run due to the lower interest rate.

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Suppose that you invest $7000 in a risky investment. at the end of the first year, the
investment has decreased by 70% of its original value. at the end of the second year,
the investment increases by 80% of the value it had at the end of the first year. your
investment consultant tells you that there must have been a 10% overall increase of
the original $7000 investment. is this an accurate statement? if not, what is your
actual percent gain or loss on the original $7000 investment. round to the nearest
percent.

Answers

The actual percent loss on the original $7000 investment is 46%. This means that the investment consultant's statement of a 10% overall increase is not accurate.

To calculate the actual percent gain or loss on the original $7000 investment, we can use the following formula:

Actual percent gain or loss = (Ending value - Beginning value) / Beginning value * 100%

At the end of the first year, the investment decreased by 70% of its original value, which means its value was only 30% of $7000, or $2100.

At the end of the second year, the investment increased by 80% of the value it had at the end of the first year. So, its value at the end of the second year was:

Value at end of second year = $2100 + 80% of $2100
Value at end of second year = $2100 + $1680
Value at end of second year = $3780

Therefore, the actual percent gain or loss on the original $7000 investment is:

Actual percent gain or loss = ($3780 - $7000) / $7000 * 100%
Actual percent gain or loss = -46%

So, the actual percent loss on the original $7000 investment is 46%.

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WALK THE PATH SHOWN WHAT IS THE DISTANCE

Answers

Answer:

D. 4π

Step-by-step explanation:

Circumference: C = 2πr = 2π(8) = 16π

The distance = 1/4 circumference (angle is 90 degrees)

=> distance = 16π/4 = 4π

Kevin works 3z hours each day from Monday to Friday. He works (4z-7) on Saturday. Kevin does not work on Sunday. Find the number of hours Kevin works in one week in terms of z

Answers

Answer:

im gooder like that

Step-by-step explanation:

3z*5=15z

15z+4z-7

19z-7

Marcus is responsible for maintaining the swimming pool in his community. He adds chemicals, when needed, to lower the pH of the pool.

-The maximum pH value allowed for the pool is 7. 8.

-The pool currently has a pH value of 6. 9.

-The pH value of the pool increases by 0. 05 per hour.

Write an inequality that can be used to determine x, the number of hours before Marcus will need to add chemicals to maintain the pH for the pool

Answers

An inequality that can be used to determine x, the number of hours before Marcus will need to add chemicals to maintain the pH for the pool would be 6.9 + 0.05x ≤ 7.8

To determine the number of hours (x) before Marcus will need to add chemicals to maintain the pool's pH, we can use an inequality with the given information.

-The maximum pH value allowed for the pool is 7.8.
-The pool currently has a pH value of 6.9.
-The pH value of the pool increases by 0.05 per hour.

The inequality for this scenario would be:

6.9 + 0.05x ≤ 7.8

This inequality states that the current pH value (6.9) plus the increase in pH per hour (0.05x) should be less than or equal to the maximum allowed pH value (7.8). This will help us determine the number of hours (x) before Marcus needs to add chemicals to maintain the pH for the pool.

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The square pyramid has a base with an area of 64 cm and a slant height of 9 cm. What is the height of the pyramid

Answers

To find the height of the square pyramid, we will use the Pythagorean theorem. Given the area of the base is 64 cm² and the slant height is 9 cm, let's first find the side length of the base.

Since it's a square, the area of the base is side length squared (s²). Therefore, s² = 64 cm². Taking the square root of both sides, we get s = 8 cm.

Now, let the height be h and use the Pythagorean theorem with the side length (8 cm) and the slant height (9 cm):

h² + (s/2)² = (slant height)²
h² + (8/2)² = 9²
h² + 4² = 81
h² + 16 = 81
h² = 65

Taking the square root of both sides:

h = √65 cm ≈ 8.06 cm

The height of the pyramid is approximately 8.06 cm.

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Lizzie came up with a divisibility test for a certain number m ≠ 1 : Break a positive integer n into two-digit chunks, starting from the ones place. (For example, the number 354764 would break into the two-digit chunks 35, 47, and 64 ) Find the alternating sum of these two-digit numbers, by adding the first number, subtracting the second, adding the third, and so on. (In our example, this alternating sum would be 35-47+64 = 52 ) Find m and show that this is indeed a divisibility test for n (by showing that n is divisible by m if and only if the result of this process is divisible by m)

Answers

If and only if the alternating sum of the two-digit chunks of a number n is divisible by m, then n is divisible by m.

What value of m makes the alternating sum of two-digit chunks of a positive integer n divisible by m?

Let's denote the two-digit chunks of n as a₁a₂, a₃a₄, ..., where a₁, a₂, a₃, a₄, ... are the digits from the ones place onward.

The alternating sum of these two-digit numbers is given by a₁a₂ - a₃a₄ + a₅a₆ - a₇a₈ + ...

We can rewrite n as (a₁a₂ × 100) + (a₃a₄ × 10) + (a₅a₆ × 1) + ...

The alternating sum expression can be written as (a₁a₂ × 100) - (a₃a₄ × 10) + (a₅a₆ × 1) - ...

If n is divisible by m, we have n ≡ 0 (mod m).

Rewriting n in terms of the alternating sum, we get (a₁a₂ × 100) - (a₃a₄ × 10) + (a₅a₆ × 1) - ... ≡ 0 (mod m).

Factoring out each two-digit chunk, we have (a₁a₂ - a₃a₄ + a₅a₆ - ...) ≡ 0 (mod m).

This implies that n is divisible by m if and only if the result of the alternating sum process is divisible by m.

m is the value that makes the alternating sum of the two-digit chunks of n a divisibility test for n.

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A radar antenna is located on a ship that is 4 kilometres from a straight shore. It is rotating at 32 revolutions per minute. How fast does the radar beam sweep along the shore when the angle between the beam and the shortest distance to the shore is Pi/4 radians?

Answers

The radar beam moves at a pace of roughly 536.47 kilometers per hour as it scans the coastline.

Let A represent the location of the radar antenna and B represent the shoreline location that is closest to A. Let C represent the radar beam's current location on the coast and Ф represent the angle between the beam and the line AB. As a result, we obtain a right triangle ABC, where AB is equal to 4 km, and BC is the length at which the radar beam sweeps along the shore.

32 rev/min(2π/60 sec) = 3.36 radians/sec. BC = r(Ф) = (4 km)(π/4) = π km.

We may calculate the radar beam's speed down the shore by multiplying these two values:

(536.47 km/hr) = 10.54 km/sec or (3.36 rad/sec)(π km).

Hence, the of sweeping is 10.54 km/sec.

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We told Gavin not to run but he did anyway he took 18 steps from second base got caught and took 7 steps back then 3 steps forward 5 steps back and 11 steps forward and 4 steps back and was tagged out halfway between second and third base how many steps is it from second base to third pls help

Answers

it is 16 steps from second base to third base .we can get this answer by

solving the logic given in the question .we need to add up the number of steps

what is add up ?

"Add up" means to calculate the total of two or more numbers or quantities by combining them together. It involves the mathematical operation of addition, which is the process of finding the sum of two or more numbers. For example, if you add up the numbers 3, 5, and 7, t

In the given question,

To find out how many steps it is from second base to third base, we need to add up the number of steps Gavin took in each direction.

Starting at second base, he took 18 steps forward to get caught, and then took 7 steps back. This leaves him 11 steps forward from second base.

Next, he took 3 more steps forward, for a total of 14 steps forward from second base. But then he took 5 steps back, leaving him 9 steps forward from second base.

Then he took 11 more steps forward, for a total of 20 steps forward from second base. But then he took 4 steps back, leaving him 16 steps forward from second base.

Finally, he was tagged out halfway between second and third base. Since he was 16 steps forward from second base, and halfway between second and third base, we can assume that third base is 16 steps away from second base.

Therefore, it is 16 steps from second base to third base.

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Indetify the mononial, binomail or trinomial

4x2 - y + oz4

Answers

The given expression is a trinomial because it consists of three terms: 4x²-y+oz⁴

A trinomial is a polynomial with three terms. It is a type of algebraic expression that consists of three monomials connected by addition or subtraction. The general form of a trinomial is:

ax^2 + bx + c

A monomial is an algebraic expression that consists of a single term. It is a polynomial with only one term. A term is a combination of a coefficient and one or more variables raised to non-negative integer exponents. The general form of a monomial is:c * xᵃ, yᵇ, zⁿ....

where 'c' represents the coefficient (a constant), and 'x', 'y', 'z', etc., represent variables, each raised to a non-negative exponent (a, b, n, etc.).

example of monomials: 5x² - This monomial has a coefficient of 5 and a single variable 'x' raised to the power of 2.

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Two containers are mathematically similar.
Their volumes are 54cm3
and 128 cm3
.
The height of the smaller container is 4.5cm.
Calculate the height of the larger container

Answers

Answer: 6cm

Step-by-step explanation: VF=(SF)^3

let x be the height of the larger container

VF1/VF2=(SF1/SF2)^3

54/128=(4.5/X)^3

rearrange to find x

x=cube root of (4.5^3*128)/54

x=6cm

< Question 7 Σ Next Use partial fraction decomposition to evaluate the integral: (198r2 + 9r – 50 dr 99r2 - 26r - 8

Answers

The value of integral of (198r² + 9r – 50 dr is (16/27)ln|11r + 2| - (2/3)ln|9r - 4| + C

Now, we can write the integrand as a sum of two fractions:

(198r² + 9r - 50) / (11r + 2)(9r - 4) = A/(11r + 2) + B/(9r - 4)

To find A and B, we need to solve for them using the method of equating coefficients:

198r² + 9r - 50 = A(9r - 4) + B(11r + 2)

Setting r = 4/9 and r = -2/11, we get two equations:

198(4/9)² + 9(4/9) - 50 = A(9(4/9) - 4) + B(11(4/9) + 2)

198(-2/11)² + 9(-2/11) - 50 = A(9(-2/11) - 4) + B(11(-2/11) + 2)

Solving these equations gives A = 16/27 and B = -2/3.

So, the integral can be written as:

∫(198r² + 9r - 50) / (11r + 2)(9r - 4) dr = ∫16/27(11r + 2)^-1 dr - ∫2/3(9r - 4)^-1 dr

Integrating each term gives:

(16/27)ln|11r + 2| - (2/3)ln|9r - 4| + C

where C is the constant of integration.

In summary, using partial fraction decomposition, we can express the given integral as a sum of two simpler integrals, which can be evaluated using the natural logarithm function.

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Find dy/dx. x =^9root (t) y = 9 - t dy/dx = _____

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To find dy/dx, we need to take the derivative of y with respect to x. On evaluating the value of dy/dx is [tex]-9t^{8/9}[/tex]

However, we are given x in terms of t. So first, we need to use the chain rule to find dx/dt:
x = [tex]t^{1/9}[/tex]
dx/dt = (1/9) * [tex]t^{-8/9}[/tex]

Now, we can use the chain rule again to find dy/dt:

y = 9 - t
dy/dt = -1

Finally, we can use the formula for the chain rule to find dy/dx:

dy/dx = (dy/dt) / (dx/dt)
dy/dx = (-1) / ((1/9) * [tex]t^{-8/9}[/tex]
dy/dx = [tex]-9t^{8/9}[/tex]

So, the final answer is dy/dx = [tex]-9t^{8/9}[/tex]
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In Exercises 1-11, calculate all four second-order partial derivatives and check that fxy = fyx. Assume the variables are restricted to a domain on which the function is defined. 1. f(x,y) = (x + y)2 2. f(x,y) = (x + y) 3. f(x,y) = 3x"y + 5xy! 4. f(x,y) = 2xy 5. f(x,y) = (x + y)ey 6. f(, y) = xe 7. f(x, y) = sin(x/y) 8. f(x,y) = x2 + y2 9. f(x, y) = 5x®y2 - 7xy? + 9x² +11 10. f(x, y) = sin(x2 + y2) 11. f(x, y) = 3 sin 2x cos 5y

Answers

For each function, all four second-order partial derivatives are f(x,y) are (x + y)2, (x + y), 3x^2y + 5xy^2, 2xy, (x + y)e^y, xe^y, sin(x/y), x^2 + y^2, 5x^3y^2 - 7xy^3 + 9x^2 +11, sin(x^2 + y^2) and 3 sin(2x) cos(5y). It is proved that f x y is equals to f y x.

f(x,y) = (x + y)2

f x x = 2, f xy = 2, f yx = 2, f y y = 2

Since f x y = fy x, the mixed partial derivatives are equal.

f(x,y) = (x + y)

f x x = 0, f x y = 1, f y x = 1, f y y = 0