The temperature at a point (x,y,z) is given by
T(x,y,z)=200e−ˣ²−⁵ʸ²−⁷ᶻ²
where T is measured in ∘C and x,y,z in meters
Find the rate of change of temperature at the point P(4,−1,4) in the direction towards the point (5,−4,5).
Answers
The rate of change of temperature at the point P(4,−1,4) in the direction towards the point (5,−4,5) is 0.
To find the rate of change of temperature at point P(4, -1, 4) in the direction towards the point (5, -4, 5), we need to calculate the gradient of the temperature function T(x, y, z) and then evaluate it at the given point.
The gradient of a function represents the rate of change of that function in different directions. In this case, we can calculate the gradient of T(x, y, z) as follows:
∇T(x, y, z) = (∂T/∂x) i + (∂T/∂y) j + (∂T/∂z) k
To calculate the partial derivatives, we differentiate each term of T(x, y, z) with respect to its respective variable:
∂T/∂x = 200e^(-x² - 5y² - 7z²) * (-2x)
∂T/∂y = 200e^(-x² - 5y² - 7z²) * (-10y)
∂T/∂z = 200e^(-x² - 5y² - 7z²) * (-14z)
Now we can substitute the coordinates of point P(4, -1, 4) into these partial derivatives:
∂T/∂x at P(4, -1, 4) = 200e^(-4² - 5(-1)² - 7(4)²) * (-2 * 4)
∂T/∂y at P(4, -1, 4) = 200e^(-4² - 5(-1)² - 7(4)²) * (-10 * -1)
∂T/∂z at P(4, -1, 4) = 200e^(-4² - 5(-1)² - 7(4)²) * (-14 * 4)
Simplifying these expressions gives us:
∂T/∂x at P(4, -1, 4) = -3200e^(-107)
∂T/∂y at P(4, -1, 4) = 2000e^(-107)
∂T/∂z at P(4, -1, 4) = -11200e^(-107)
Now, to find the rate of change of temperature at point P in the direction towards the point (5, -4, 5), we can use the direction vector from P to (5, -4, 5), which is:
v = (5 - 4)i + (-4 - (-1))j + (5 - 4)k
= i - 3j + k
The rate of change of temperature in the direction of vector v is given by the dot product of the gradient and the unit vector in the direction of v:
Rate of change = ∇T(x, y, z) · (v/|v|)
To calculate the dot product, we need to normalize the vector v:
|v| = √(1² + (-3)² + 1²)
= √(1 + 9 + 1)
= √11
Normalized vector v/|v| is given by:
v/|v| = (1/√11)i + (-3/√11)j + (1/√11)k
Finally, we can calculate the rate of change:
Rate of change = ∇T(x, y, z) · (v/|v|)
= (-3200e^(-107)) * (1/√11) + (2000e^(-107)) * (-3/√11) + (-11200e^(-107)) * (1/√11)
= 0
Since, the value of e^(-107) = 0.
Therefore, rate of change = 0.
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Related Questions
If a cook works 38 hours in one week and 36 hours the next week, what is the total number of hours worked by the cook in the two-week period? If the cook gets paid $10.50 per hour, what will be his gross pay (before taxes) for the two-week period?
Answers
435
Step-by-step explanation:
38+36 x10.50=435 gross pay
In trapezoid ABCD(AB ∥CD), point M∈AD, so that AM:MD=3:5. Line l ∥ AB and passes through point M to intersect diagonal AC and leg BC at points P and N ,respectively. Find AC:PC.
there are no pictures. Will give brainlest.
Answers
Step by step explanation
Solve for y.
x + a=yb
O y=x+a-b
O y= (x+a)
b
O y= (x+a)
b
Answers
Answer:
The answer is the 2nd bullet point (B)
What is the slope of the line that passes through the points (8, -6)(5,−1)? Write your answer in simplest form.
Answers
Answer:
\(-\frac{5}{3}\)
Step-by-step explanation:
\(m (slope)=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
\(m=\frac{-1-(-6)}{5-8}\)
\(m=-\frac{5}{3}\)
Integrated circuits from a certain factory pass quality test with probability ,8,p=,8. The outcomes of tests are mutually independent. Use The CTL to estimate the probability of finding at most of 50 acceptable circuits in a batch of 60 .
Answers
The estimated probability of finding at most 50 acceptable circuits in a batch of 60 is approximately 0.6591.
What is the estimated probability of obtaining no more than 50 acceptable circuits in a batch of 60, given a pass probability of 0.8 and independent outcomes?To estimate the probability of finding at most 50 acceptable circuits in a batch of 60 from a certain factory, where the probability of passing the quality test is (p = 0.8) and the outcomes of the tests are mutually independent, we can use the Central Limit Theorem (CLT).
The CLT states that for a large enough sample size, the distribution of the sample mean approaches a normal distribution, regardless of the shape of the population distribution.
Let's denote (X) as the number of acceptable circuits in a batch of 60. Since each circuit passes the test with a probability of 0.8, we can model (X) as a binomial random variable with parameters (n = 60) and (p = 0.8).
To estimate the probability of finding at most 50 acceptable circuits, we can calculate the cumulative probability using the normal approximation to the binomial distribution.
Since the sample size is large \((\(n = 60\))\), we can approximate the distribution of (X) as a normal distribution with mean \(\(\mu = np = 60 \times 0.8 = 48\)\) and standard deviation \(\(\sigma = \sqrt{np(1-p)}\) = \(\sqrt{60 \times 0.8 \times 0.2} \approx 4.90\).\)
Now, we want to find the probability of\(\(P(X \leq 50)\)\). We can standardize the value using the z-score:
\(\[P(X \leq 50) = P\left(\frac{X - \mu}{\sigma} \leq \frac{50 - 48}{4.90}\right) = P(Z \leq 0.41)\]\)
Using the standard normal distribution table or calculator, we can find that \(\(P(Z \leq 0.41) \approx 0.6591\).\)
Therefore, the estimated probability of finding at most 50 acceptable circuits in a batch of 60 is approximately 0.6591.
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Please help!!!!!!!!!!
Answers
if 15 x 1.8 = 27 then
45 x 1.8 = 81
so answer: B) 81
Every time Luisa bakes a batch of brownies, she uses 12 cup of chocolate. If she has 32 cup(s) of chocolate remaining, how many batches of brownies can Luisa make? Input your answer as a whole number.
Answers
This question was not properly written
Complete Question
Every time Luisa bakes a batch of brownies she uses 1/2 cup of chocolate if she has 3/2 cups of chocolate remaining how many batches of brownies can Luisa make
Answer:
3 batches of brownies from 3/2 cups of chocolate left.
Step-by-step explanation:
From the question, we know that:
1/2 cups of chocolate = 1 batch of brownies
3/2 cups of chocolate = x
Cross Multiply
1/2 × x = 3/2
x = 3/2/1/2
x = 3/2 × 2/1
x = 3 batches.
Approximately as a whole number , he can make 3 batches of brownies from 3/2 cups of chocolate left
A shopkeeper had 410 tangerines. He put some of them into 15 cartons containing
12 tangerines each. He then put the rest into 14 cartons, each containing the same
number of tangerines.
Answers
Answer:
15*14 + 14*x = 410
x = 200/14 = 14 4/14 <===== 14*14 + 4 = 200
Step-by-step explanation:
In your own words, explain the relationship between the graph of a function and the graph of function's derivative.
Answers
Answer:
The derivative of a function in a point is the slope of the graph in that point, so the graph of the function's derivative shows the slope of the function at any point. It is useful because we can say if the function is increasing or decreasing using the graph of the function's derivative. Where the derivative is positive, the graph of the function is increasing and when the derivative is negative, the graph of the function is decreasing.
Using x for the length of each side in an equilateral triangle, express its area in terms of x.
Answers
The solution is:
1. Height of the equilateral triangle is: √3 units
2. Area of the equilateral triangle = √3 units²
3. Area = √3/4 x², when each side is x.
We have,
A triangle that has all its three sides equal in length, is referred to as an equilateral triangle.
1. Given the each side measures 2 units, and h is the height, applying the Pythagorean theorem, we would have:
h = √(2² - 1²)
h = √(4 - 1)
h = √3
2. Area of the equilateral triangle = 1/2bh = 1/2(2)(√3)
Area of the equilateral triangle = √3 units²
3. If x is the side length of the equilateral triangle, we would have:
height (h) = √[x² - (x/2)²] = √[x² - x²/4] = √(3x²/4) = √3/2x
Area = 1/2bh = 1/2(x)(√3/2x) = √3/4 x²
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complete question:
Here is an equilateral triangle. The length of each side is units. A height is drawn. In an equilateral triangle, a line drawn from one corner to the center of the opposite side represents the height. 1. Find the exact height. 2. Find the area of the equilateral triangle. 3. (Challenge) Using for the length of each side in an equilateral triangle, express its area in terms of .
a 9-ft roll of wrapping paper costs $6.48, what is the price per inch?
Answers
Answer: $0.06 per inch
Concept:
When encountering questions that ask for unit price, follow the statements to get the answer. For example, this question asks the price per inch. Therefore, in the actual answer, use the total price given divided by the number of inches.
Solve:
Price = $6.48
Inch = 9 × 12 = 108 in
1 ft = 12 inchPrice / Inch = 6.48 / 108 = $0.06 per inch
Hope this helps!! :)
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Find the center of mass of cone of uniform density that has a radius R at the base, height h, and mass M. Let the origin be at the center of the base of the cone and have +z going through the cone vertex.
Answers
To find the center of mass of a cone of uniform density with radius R at the base, height h, and mass M, we need to use the formula:
x_cm = (1/M)∫∫∫xρdV
y_cm = (1/M)∫∫∫yρdV
z_cm = (1/M)∫∫∫zρdV
where x_cm, y_cm, and z_cm are the coordinates of the center of mass, ρ is the density, and V is the volume of the cone.
We can simplify the integral by using cylindrical coordinates, where the density is constant and equal to M/V, and the limits of integration are:
0 ≤ r ≤ R
0 ≤ θ ≤ 2π
0 ≤ z ≤ h(r/R)
Thus, the center of mass of the cone is:
x_cm = 0
y_cm = 0
z_cm = (3h/4)(r/R)^2
Therefore, the center of mass of the cone is located at (0, 0, (3h/4)(r/R)^2) with respect to the origin at the center of the base of the cone and +z going through the cone vertex.
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It takes the earth 24 h to complete a full rotation. It takes Mercury approximately 58 days, 15 h, and 30 min to complete a full rotation. How many hours does it take Mercury to complete a full rotation? Show your work using the correct conversion factors.
Answers
Answer: 1407.5 hours
Step-by-step explanation:
To solve this problem, we first need to convert the time it takes Mercury to complete a full rotation into hours. We can do this by multiplying the number of days by 24 to get the number of hours, and then adding the number of hours and minutes to find the total number of hours. This gives us:
(58 days * 24 h/day) + 15 h + (30 min * 1 h/60 min)
1392 h + 15 h + 0.5 h
1407.5 h
Therefore, it takes Mercury approximately 1407.5 hours to complete a full rotation. This can also be written as 1407.5 hours/rotation or 1407.5 h/rotation.
Plz help will give brainalist. Please try...
Answers
Its 382.98 the answer
1:3, 9:28, 4:12, 2:6, 6:18 which one is the odd one out?
Answers
Answer:
9:28
Step-by-step explanation:
because the rest are equivalent to 1:3.
please mark as brainliest
Answer:
9:28 is the odd one
Evaluate the expression shown below and write your answer as a fraction in simplest form(1/10-2/3) ÷ 0.5
Answers
Answer:
\(\frac{-17}{15}\)
Step-by-step explanation:
When solving this problem you must follow PEMDAS, so parentheses should go first. To solve 1/10 - 2/3, first, find the common denominator. In this case, that is 30. Then, subtract, \(\frac{3}{30} - \frac{20}{30}\). This equals -17/30. Finally, divide by 0.5. To do this remember that dividing by half is the same as multiplying by 2. Doing \(\frac{-17}{30} *2\) equals \(\frac{-17}{15}\).
Solve for x trigonometry
Answers
Answer:
Set your calculator to degree mode.
tan(37°) = x/8
x = 8tan(37°) = 6.03
For the given triangle the required value of the x is approximately 6.02 units.
Use the concept of triangle defined as:
A triangle is a three-sided polygon, which has three vertices and three angles which has a sum of 180 degrees.
In the given triangle:
One angle is 37°
Adjacent side = 8 cm
Perpendicular = x
Since we know that,
Tan θ = opposite/adjacent
Therefore,
Tan 37° = x/8
Since Tan 37° ≈ 0.753
Then, 0.753 ≈ x/8
x ≈ 6.02 units
Hence,
The required value of x is approximately 60.2 units.
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how many strings of 6 lowercase english letters are there that start and end with the letters cu (in that order) if letters can be repeated?
Answers
There are 456,976 strings of 6 lowercase English letters that start and end with the letters "cu" if letters can be repeated.
We are asked to find the number of strings of 6 lowercase English letters that start and end with the letters "cu" (in that order) if letters can be repeated.
The problem is simplified by the requirement that the string start and end with "cu". Since these two letters are fixed, we only need to consider the four letters in between. We can treat each of these four letters as an independent choice from a set of 26 lowercase English letters (since repetition is allowed). Thus, there are 26 choices for each of the four positions, giving us a total of 26^4 possible strings for the four letters in the middle.
Since there is only one possible choice for the first letter ("c") and one possible choice for the last letter ("u"), the total number of strings that start and end with "cu" and have any four letters in between is simply the product of the number of choices for the middle four letters and the fixed first and last letters, which is:
26^4 = 456976
So there are 456,976 strings of 6 lowercase English letters that start and end with the letters "cu" if letters can be repeated.
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The table shows the balances for 3 bank accounts.
Use this information to complete the statements below.
Answers
A $950 deposit for 18 years compounded at
an annual interest rate of 2.21%
Answers
Answer: 1408.01$
Step-by-step explanation: Use the compound interest formula P*(1+r)^n Where P is the initial value, r is the interest rate, and n is the number of periods
Given the formula for an exponential function, is it possible to determine whether the function grows or decays exponentially just by looking at the formula?
Answers
yes, actually it is possible. The mathematical expression f(x)=exp or ex denotes the exponential function.
What is meant exponential function?The mathematical expression f(x)=exp or ex denotes the exponential function. The word, unless specifically stated differently, normally refers to the positive-valued function of a real variable, though it can be extended to the complex numbers or adapted to other mathematical objects like matrices or Lie algebras. When the input variable x appears as an exponent in the formula f(x) = axe, an exponential function is indicated. The exponential curve is influenced by both the exponential function and the value of x.
One of the key mathematical operations is the exponential function (though it would have to admit that the linear function ranks even higher in importance). We allow the exponent to be the independent variable to create an exponential function. The formula f(x)=2x is a straightforward example.
Hence, yes, actually it is possible.
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As the softball sails through the air, a defensive player raises her hand and catches the ball in her glove on its way down. the height of her raised glove is 6.5 feet. how long does it take from the time the softball was struck by the bat for the player to catch it? round your answer to the nearest hundredth of a second, and express with the appropriate unit. (hints: is 6.5 a value oft or( )h t ? consequently, will our answer be a value oft or( )h t ?)184vv
Answers
It take from the time the softball was struck by the bat for the player to catch it is 3.9 sec
What is Time?
Time is the ongoing progression of existence and events from the past through the present and into the future, which appears to be an irrevocable succession.
Given,
The height of the ball when its caught is h(t) = 6.5feet
h(t) = -16t² +63t +4.1
6.5 = -16t² +63t +4.1
16t² - 63t +6.5 - 4.1 = 0
16t² - 63t +2.4= 0
Solving for t
t = -(-63) ±√63² - 4(16)(2.4)/2(16)
t = 63 ±√3969-153.6/32
t = 63 ±√3815.4/32
t = 63 ±61.76/32
t = 63 - 61.76/32
or
t = 63 + 61.76/32
t = 0.038
t = 3.9
t can't be 0.038 since at t = 0.038 the ball reaches 6.5 feet while going up
but while coming down it reaches 6.5 feets at t = 3.9sec
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17. On a map, 1 inch = 20 miles. If the road is 4.3 inches long on the map, how many miles is the actual road?
18. If you travel 450 miles on 15 gallons of gas, how many gallons will you need to travel 1530 miles?
19. Sam's ratio of blue marbles to black marbles is 3:5. How many blue marbles does Sam have if he has 45 black
marbles?
20. Mandi is carrying a 10 liter jug of sports drink that weighs 15 kg. She wants to know how many kilograms a 2-liter
jug of sports drinks would weigh.
Answers
Answer:
17. 86 miles
Step-by-step explanation:
What is the circumference of a circle with a diameter of 12 feet? (use 3.14 for pi)
36 ft.
37.68 ft.
36.26 ft.
38 ft.
Answers
Answer:
37.68ft
Step-by-step explanation:
Answer:
37.68 ft
Step-by-step explanation:
The circumference of a circle is given by
C = pi *d where d is the diameter
C = 3.14 * 12
C =37.68 ft
A map of Rockbridge County has a scale factor of
2.5 in = 5 mi
if the actual distance from Fairfield to Lexington is 12 miles what is the distance on the map?
(plzzz show work I'm doing a test rn thats half my grade plzzzzzz)
Answers
Answer: 6 inch
Step-by-step explanation:
From the question, we are informed that a map of Rockbridge County has a scale factor of 2.5 in = 5 miles.
When the actual distance from Fairfield to Lexington is 12 miles, the distance on the map will be:
= 12/5 × 2.5
= 2.4 × 2.5
= 6 inch
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Enlarge shape A by scale factor 2 with centre of enlargement (-2, -5).
What are the coordinates of the vertices of the image?
Answers
The coordinates of the image of the shape A with vertices at (2, 3), (4, 7), (8, 3), and (-2, -5) enlarged by a scale factor 2 with centre of enlargement at (-2, -5) are (0, -1), (2, 1), (4, -1), and (-4, -9).
Given a shape A with vertices at (2, 3), (4, 7), (8, 3), and (-2, -5) and a scale factor of 2 with a centre of enlargement at (-2, -5), the coordinates of the vertices of the image are (0, -1), (2, 1), (4, -1), and (-4, -9).
We can calculate the new coordinates of the vertices by multiplying the scale factor with the differences between the coordinates of the vertices and the coordinates of the centre of enlargement, then adding the coordinates of the centre of enlargement to the product.
For example, the x and y coordinates of the first vertex (2, 3) are (2, 3) and the centre of enlargement is (-2, -5). So, the difference between the two is (2 - (-2), 3 - (-5)) = (4, 8). Multiplying the scale factor 2 with the differences gives (4 * 2, 8 * 2) = (8, 16). Adding the centre of enlargement to the product gives (8 + (-2), 16 + (-5)) = (6, 11). Therefore, the coordinates of the image of the first vertex is (6, 11).
Similarly, for the second, third, and fourth vertices, the coordinates of the image are (4, 5), (8, 1), and (-4, -9) respectively.
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a histogram with the hills to the far left means the image is overexposed.
Answers
The statement "A histogram with the hills to the far left means the image is overexposed" is not accurate. In fact, a histogram with the hills to the far left indicates that the image is underexposed, not overexposed.
A histogram is a graphical representation of the distribution of pixel values in an image. It displays the frequency of occurrence of different tonal values. The horizontal axis represents the tonal values, while the vertical axis represents the frequency or number of pixels.
When the hills of the histogram are concentrated towards the left side, it means that a significant portion of the image has darker or lower tonal values. This indicates an underexposed image, where the exposure settings were insufficient to capture enough light, resulting in a darker overall appearance.
Conversely, an overexposed image would have the hills of the histogram concentrated towards the right side, indicating that a significant portion of the image has brighter or higher tonal values.
Therefore, the correct interpretation is that a histogram with the hills to the far left means the image is underexposed, not overexposed.
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Leah would like to earn at least $120 per month. She babysits for $5 per hour and works at an ice cream shop for $8 per hour. Leah cannot work more than a total of 20 hours per month. Let x represent the number of hours Leah babysits and let y represent the number of hours Leah works at the ice cream shop.
Which pair (x, y) represents hours that Leah could work to meet the given conditions? Select all that apply.
A.
(2.5, 15)
B.
(5, 20)
C.
(7.5, 12.5)
D.
(17.5, 5)
E.
(19, 1)
Answers
Answer:
A C
Step-by-step explanation:
Answer:
Step-by-step explanation:
Leah wants to earn at least 120 dollars a month and she has two job options. This is written as
5x + 8y > 120
She also can't work more than 20 hours, leading us to the second equation
x + y = 20
First we solve for one variable
y = 20 - x
then substitute it into the first equation
5x + 8(20 - x) = 120
Solve for x
5x + 160 - 8x = 120
3x = 40
x = 40/3
Then we solve for y
y = 20 - x
y = 20/3
x < 40/3
y > 20/3
B and D will not work because their x+y is greater than 20.
E will also not work because the value is less than 120
Choose the inequality that represents the following graph
Answers
Answer:
x<5
Step-by-step explanation:
The graph is from 5(not included) to negative infinity
Answer:
Answer is A!
Step-by-step explanation:
Basically, If the dot is NOT filled in, them it X will be ANY of the numbers (aside from the number the dot is on) on the blue arrow.
Your problem is asking you if numbers from 4 to -5 are bigger or smaller than 5.
The answers that have ≤ or ≥ mean that is can be either (below or higher depending on the direction the ≤ or ≥ is facing) or equal to the number on the dot. Hope this helps!
Disk drives have been getting larger. Their capacity is now often given in terabytes (TB) where 1 TB = 1000 gigabytes, or about a trillion bytes. A survey of prices for external disk drives found the data shown in chart. For this data, we want to predict Price from Capacity. A). Find the slope estimate b1. _________B). Find the intercept b0. ___________C). Write equation that predicts Price from Capacity - price= ______ + (_____) capacityD). What would you predict price of a 20.0 TB disk? _______
Answers
To compute the slope, intercept, and equation to predict the Price, we will use the formulae below:
For the equation
\(\bar{y}=b(\bar{x})+a\)For the slope
\(b=r(\frac{S_y}{S_x})\)For the intercept
\(a=\bar{y}-b\bar{x}\)In this case, we have the following data given:
\(\begin{gathered} \bar{x}=7.611,S_x=9.85,r=0.987 \\ \bar{y}=784.173,S_y=1419.89 \end{gathered}\)Part A
Upon substituting the above into the formula, we will have
\(b=0.987(\frac{1419.89}{9.85})=141.41\)Hence slope = 141.41
Part B
\(\begin{gathered} a=\bar{y}-\bar{bx} \\ a=784.173-141.41(7.611) \\ a=-292.099 \end{gathered}\)Hence, Intercept =-292.099
PART C
The equation that can be used to predict Price from Capacity
\(\begin{gathered} \text{Price}=\text{intercept}+(\text{slope)capacity} \\ \text{Price}=-292.099+(141.41)\text{capacity} \end{gathered}\)Therefore, the equation is
Price=-292.099+(141.41)capacity
Part D
To predict the price of a 20.0TB disk, we will simply set the capacity= to 20 and then simplify
\(\begin{gathered} P_{20TB}=-292.099+141.41(20) \\ P_{20TB}=2536.10 \end{gathered}\)Hence, the predicted price is=2536.10