this is 8th grade math and i dont understand it

The directional derivative of the function f(x, y, z) at point P0(2, 3, 4) in the direction of a = 3i - 4j + 5k is 648/5.

To find the derivative of the function at the given point in the direction of a, we need to use directional derivative formula. The directional derivative of a function f(x, y, z) at a point P0 in the direction of a = ai + bj + ck is given by the dot product of the gradient of the function f(x, y, z) at P0 and the unit vector in the direction of a.

So, the formula for the directional derivative of a function f(x, y, z) at a point P0(x0, y0, z0) in the direction of a = ai + bj + ck is given by Daf = ∇f(x, y, z)·(a/|a|)where, ∇f(x, y, z) is the gradient vector of the function f(x, y, z) and (a/|a|) is the unit vector in the direction of a.

Let's solve the given problem using the above formula.5) f(x, y, z) = x²y³z, P0(2, 3, 4), a = 3i - 4j + 5k We need to find the directional derivative of the function f(x, y, z) at point P0(2, 3, 4) in the direction of a = 3i - 4j + 5k. We have to find the gradient of the function f(x, y, z) first. So,∇f(x, y, z) = (df/dx)i + (df/dy)j + (df/dz)k Here, f(x, y, z) = x²y³z∴ df/dx = 2xy³z∴ df/dy = 3x²y²z∴ df/dz = x²y³Putting these values in the gradient formula, we get,∇f(x, y, z) = (2xy³z)i + (3x²y²z)j + (x²y³)k So, the gradient of the function f(x, y, z) is ∇f(x, y, z) = (2xy³z)i + (3x²y²z)j + (x²y³)k

Now, we need to find the unit vector in the direction of a = 3i - 4j + 5k.

The magnitude of a is,|a| = √(3² + (-4)² + 5²) = √50 = 5√2So, the unit vector in the direction of a is,(a/|a|) = [(3/5) i - (4/5) j + (5/5) k] = (3/5) i - (4/5) j + k Therefore, the directional derivative of f(x, y, z) at point P0(2, 3, 4) in the direction of a = 3i - 4j + 5k is given by,Daf = ∇f(x, y, z)·(a/|a|)= [(2xy³z)i + (3x²y²z)j + (x²y³)k] · [(3/5) i - (4/5) j + k]= (6xy³z/5) - (12x²y²z/5) + (3x²y³/5)Now, we need to find the directional derivative of the function f(x, y, z) at point P0(2, 3, 4) in the direction of a = 3i - 4j + 5k.

So, we have to put the values of x, y, and z in the expression we got for Daf. We have, x = 2, y = 3, and z = 4.So, the directional derivative of f(x, y, z) at point P0(2, 3, 4) in the direction of a = 3i - 4j + 5k is, Daf = (6(2)(3³)(4)/5) - (12(2²)(3²)(4)/5) + (3(2²)(3³)/5)= 864/5 - 288/5 + 72/5= 648/5 Therefore, the directional derivative of the function f(x, y, z) at the point P0(2, 3, 4) in the direction of a = 3i - 4j + 5k is 648/5.

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Answer:

you have the correct answers

Step-by-step explanation:

you already have the right answers

A major disadvantage of electron microscopes compared to light microscopes is that they cannot be used to view live specimens. Therefore, option B is the correct answer.

A major disadvantage of electron microscopes compared to light microscopes is that they cannot be used to view live specimens. This is because the electron microscope requires a vacuum environment to function properly, which would kill any live specimen. Additionally, electron microscopes can only see surface details and do not have a very high power of resolution. Lastly, electron microscopes can only be used by doctors or trained technicians, so they are not as widely available as light microscopes.

Therefore, option B is the correct answer.

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A disadvantage of electron microscopes compared to light microscopes is that they cannot be used to view live specimens (option b). Electron microscopes use a beam of electrons to create an image, which requires a vacuum environment. This means that living organisms cannot survive in the vacuum and therefore cannot be observed with electron microscopes.

I hope this helped! :)

a) U is subspace of R^3.

b) The set {(3, -2, 0), (0, 1/2, 1)} is a basis for U.

c) 2.

a) To show that U is a subspace of R^3, we need to verify the following three conditions:

i) The zero vector (0, 0, 0) is in U.

ii) U is closed under addition.

iii) U is closed under scalar multiplication.

i) The zero vector is in U since 0 + 2(0) - 3(0) = 0.

ii) Let (x1, y1, z1) and (x2, y2, z2) be two vectors in U. Then we have:

x1 + 2y1 - 3z1 = 0 (by definition of U)

x2 + 2y2 - 3z2 = 0 (by definition of U)

Adding these two equations, we get:

(x1 + x2) + 2(y1 + y2) - 3(z1 + z2) = 0

which shows that the sum (x1 + x2, y1 + y2, z1 + z2) is also in U. Therefore, U is closed under addition.

iii) Let (x, y, z) be a vector in U, and let c be a scalar. Then we have:

x + 2y - 3z = 0 (by definition of U)

Multiplying both sides of this equation by c, we get:

cx + 2cy - 3cz = 0

which shows that the vector (cx, cy, cz) is also in U. Therefore, U is closed under scalar multiplication.

Since U satisfies all three conditions, it is a subspace of R^3.

b) To find a basis for U, we can start by setting z = t (where t is an arbitrary parameter), and then solving for x and y in terms of t. From the equation x + 2y - 3z = 0, we have:

x = 3z - 2y

y = (x - 3z)/2

Substituting z = t into these equations, we get:

x = 3t - 2y

y = (x - 3t)/2

Now, we can express any vector in U as a linear combination of two vectors of the form (3, -2, 0) and (0, 1/2, 1), since:

(x, y, z) = x(3, -2, 0) + y(0, 1/2, 1) = (3x, -2x + (1/2)y, y + z)

Therefore, the set {(3, -2, 0), (0, 1/2, 1)} is a basis for U.

c) Since the basis for U has two elements, the dimension of U is 2.

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Answer:

the answer is (a) 28.

Step-by-step explanation:

The given situation can be represented by a linear equation of the form:

Cost = mx + b

where "m" is the slope (the rate of change of cost with respect to time), "x" is the number of hours of the tour, and "b" is the y-intercept (the cost when x = 0).

In this case, the y-intercept is $60, which represents the one-time fee. The cost increases by $28 per hour, so the slope is:

m = $28/hour

Therefore, the answer is (a) 28.

The length of the given rectangle is x + 5 meters.

Given:

To find: Length of the rectangle

Finding:

Let 'a' be the length of the rectangle.

We know that area of a rectangle = length * width

On substituting the given values in the above formula, we get:

=>  x² + 8x + 15 = a (x + 3)

=> a = \(\frac{x^2+8x + 15}{x+3}\)

=> a = \(\frac{(x+5)(x+3)}{x+3}\)

=> a = x + 5

Hence, The length of the given rectangle is x + 5 meters.

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The value of a1 in a geometric series for which Sn=21825, an=125, r=15 will be 5/9.

The formula for the nth term in a geometric series is given by:

an = a1 * r^(n-1)

Where a1 is the first term, r is the common ratio, and n is the number of terms.

We are given Sn, an and r, and we can use the formula for the sum of a geometric series to solve for n:

Sn = a1 * (1 - r^n) / (1 - r)

Substituting the given values, we get:

21825 = a1 * (1 - 15^n) / (-14)

Multiplying both sides by -14, we get:

-304350 = a1 * (1 - 15^n)

Since we are given that an = 125, we can substitute this into the formula for an and solve for n:

125 = a1 * 15^(n-1)

Substituting a1 = -304350 / (1 - 15^n) from the equation above, we get:

125 = -304350 / (1 - 15^n) * 15^(n-1)

Dividing both sides by 15^(n-1), we get:

125 * 15^(1-n) = -304350 / (1 - 15^n)

Multiplying both sides by (1 - 15^n), we get:

125 * 15^(1-n) * (1 - 15^n) = -304350

Expanding the right-hand side, we get:

125 * (15 - 15^n) = -304350

Dividing both sides by 125, we get:

15 - 15^n = -2434.8

Adding 15^n to both sides, we get:

15^n = 15 - (-2434.8) = 2449.8

Taking the natural logarithm of both sides, we get:

n * log(15) = log(2449.8)

Dividing both sides by log(15), we get:

n = log(2449.8) / log(15)

Since n is the number of terms, it must be a positive integer. The closest positive integer to this value is 3, so n = 3.

Finally, we can use the formula for an to find a1:

an = a1 * r^(n-1)

Substituting the given values and n = 3, we get:

125 = a1 * 15^(3-1)

Dividing both sides by 15^(3-1), we get:

125 / 15^2 = a1

Calculating, we get:

a1 = 125 / 15^2

= 125 / 225

= 5/9.

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Answer:

Lin originally had 4 bags * the same amount of apples per bag = 4 * x apples.

After eating 1 apple from each bag, she was left with 4 * x - 4 apples = 28 apples.

Therefore, each bag originally contained x = 28 + 4 = 32 apples.

Answer:

Step-by-step explanation:25

Answer:

Step-by-step explanation:

b

Answer:

Option 2: 67.014

Step-by-step explanation:

Answer:

4096

Step-by-step explanation:

Math

The highest score for the given distribution is 25.

This is because the distribution is divided into ranges, with the first range being 20-25, the second range being 15-19, the third range being 10-14, and the fourth range being 5-9.

The highest score in each range is the number on the right side of the dash.

Therefore, the highest score in the first range is 25, the highest score in the second range is 19, the highest score in the third range is 14, and the highest score in the fourth range is 9.

Since 25 is the highest score among all of the ranges, it is the highest score for the entire distribution.

The correct answer is c) 25.

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Step-by-step explanation:

v = u + at

v-u = at

a = (v -u)/t

When t = 4, u=10, v=50

a = (50-10)/4

a= 40/4

a = 10

The complete factorization of the polynomial x³ + 2x² -x - 2 is equal to (x + 2)(x + 1)(x - 1)

An equation is an expression that shows how numbers and variables are related to each other using mathematical operators.

A polynomial is an expression consisting of coefficients and variables forming an equation.

Given the polynomial:

x³ + 2x² -x - 2

Factorizing:

= (x + 2)(x² - 1)

= (x + 2)(x + 1)(x - 1)

The polynomial x³ + 2x² -x - 2 is equal to (x + 2)(x + 1)(x - 1)

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Using the law of cosines, it is found that the length of side AB is of AB = 13.

The law of cosines states that we can find the angle C of a triangle as follows:

\(c^2 = a^2 + b^2 - 2ab\cos{C}\)

in which:

For this problem, the parameters are:

C = 120, a = 8, b = 7.

Hence:

\(c^2 = a^2 + b^2 - 2ab\cos{C}\)

\(c^2 = 8^2 + 7^2 - 2(8)(7)\cos{120^\circ}\)

\(c^2 = 169\)

\(c = \sqrt{169}\)

c = 13.

Hence AB = 13.

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4x-6+3x+6=

7x

THE ANSWER IS "A"

Answer:

A 7x

Step-by-step explanation:

(4x - 6) + (3x + 6)

4x - 6 + 3x + 6

7x

Step-by-step explanation:

Since the initial fee is fixed, we can assume it is a constant and the succeeding mile fee is the variable as it changes for every mile covered.

Let x = distance (mile)

let y = total charge (dollars)

Yellowcab company equation for cost calculation;  3.50 + 2.5x = y

laxicab company equation for cost calculation;  4.54+2.3x = z

the two companies charge the same amount when y = z

; 3.50 + 2.5x = 4.54+2.3x

x = 5.2 miles

COST CALCULATION

;3.50 + 2.5x = ?

if x = 5.2;

; 3.50 + 2.5(5.2) =$16.50

OR

; 4.54+2.3x = ?

; 4.54 + 2.3(5.2) = $16.50

Answer: The volume of a cone is given by the formula: V = (1/3) * π * r^2 * h

Where:

V is the volume

π is 3.14

r is the radius

h is the height

So in this case, using the given values:

V = (1/3) * 3.14 * (25^2) * 18

V = (1/3) * 3.14 * 625 * 18

V = (625 * 18 * 3.14) / 3

V = 35,325 ft^3

So the answer is (B) 35,325 ft3

Step-by-step explanation:

Answer:

80

Step-by-step explanation:

Answer: I think it’s true

Step-by-step explanation:

The length of side b is 56m, angle A is 45°, and angle C is 67°.

In the given triangle with side lengths a = 74m, b ≈ 56m, and c = 36m, the length of side b is approximately 56m.

To solve the triangle, we can use the Law of Cosines and the fact that the sum of angles in a triangle is 180 degrees. Given angle B = 67°51', we have:

Using the Law of Cosines, we have:

b² = a² + c² - 2ac * cos(B)

Substituting the known values:

b² = 74² + 36² - 2 * 74 * 36 * cos(67°51')

Calculating the value of b:

b ≈ √(74² + 36² - 2 * 74 * 36 * cos(67°51'))

b ≈ 55.92m (rounded to the nearest whole number, b ≈ 56m)

Using the Law of Cosines again, we have:

cos(A) = (b² + c² - a²) / (2 * b * c)

Substituting the known values:

cos(A) = (56² + 36² - 74²) / (2 * 56 * 36)

Calculating the value of A:

A = cos⁻¹((56² + 36² - 74²) / (2 * 56 * 36))

A ≈ 45° (rounded to the nearest whole number)

Using the fact that the sum of angles in a triangle is 180 degrees:

C = 180° - A - B

Substituting the known values:

C ≈ 180° - 45° - 67°51'

Calculating the value of C:

C ≈ 67°9' (rounded to the nearest whole number, C ≈ 67°)

Therefore, in the given triangle, the length of side b is approximately 56m, angle A is approximately 45°, and angle C is approximately 67°.

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The equation of the line perpendicular to -x + 2 = y that intersects the line at (-1.5, 3.5) is y + x/2 = 7/2. Here's how to solve it: When writing the equation of a line perpendicular to another line, keep in mind that the slopes of the lines are negative reciprocals of each other.

The equation of the line perpendicular to -x + 2 = y that intersects the line at (-1.5, 3.5) is y + x/2 = 7/2. Here's how to solve it: When writing the equation of a line perpendicular to another line, keep in mind that the slopes of the lines are negative reciprocals of each other. This means that if one line has a slope of m, then a line perpendicular to it has a slope of -1/m. As a result, the slope of the line in the problem is 1/2, which means that the slope of the line perpendicular to it is -2. We'll use point-slope form to determine the equation of the perpendicular line because we have a point (-1.5, 3.5) and a slope (-2).

y - y1 = m(x - x1)

is the point-slope equation, where m is the slope and (x1, y1) is a point on the line that we're using. We can substitute in the values we have to get: y - 3.5 = -2(x + 1.5)

We can simplify this equation: y - 3.5 = -2x - 3y + x/2 = 7/2y + x/2 = 7/2

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The missing measures are:

BCDE is a rhombus where ,

BC = 28 and BD = 32

BC is one of the sides of the rhombus, thus, all the sides (CD, DE, EB) will be 28 cm.

So, CD = 28cm

Now, BD = 32 which is the diagonal of the rhombus.

The diagonals bisect each other at a 90-degree angle.

Hence, FD = 1/2 x BD = 1/2x32 = 16 cm.

So, FD = 16 cm

To find EF, we can consider ΔEFD , which is right-angled at F, FD = 16cm and ED = 28cm  .

Thus, from Pythagoras theorem,

ED² = EF² + FD²

28² = EF² + 16²

EF = 22.97

EF = 23 cm

So, EF = 23 cm

EC would be two times EF( since diagonals bisect each other)

Therefore, EC = 2x23 = 46 cm

So, EC = 46 cm

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Complete question:

If each quadrilateral is a rhombus, find the missing measures

BC = 28 and BD = 32

Answer:

Option B, 187.5 cm³

Step-by-step explanation:

Volume of the triangular prism,

base area × height

= (10×7.5/2)×5

= 187.5 cm³