A company has found that the daily demand x for its boxes of chocolates is inversely proportional to the price p. When the price is $6, the demand 1,800 boxes. Approximate the demand, in boxes, when the price is decreased to $2.25.

The first partial derivatives of f(x,y) at the point (3,1) are:
∂f/∂x(3,1) = 3
∂f/∂y(3,1) = -12

To find the first partial derivatives of f(x,y) at the point (3,1), we need to find the partial derivative with respect to x and y, respectively, and then substitute x=3 and y=1.

So, let's begin with the partial derivative with respect to x:
∂f/∂x = 3 - 0  (since the derivative of 3x with respect to x is 3, and the derivative of 4y with respect to x is 0)

Now, we can substitute x=3 and y=1 into this expression:
∂f/∂x(3,1) = 3 - 0 = 3

So, the partial derivative of f(x,y) with respect to x at the point (3,1) is 3.

Next, let's find the partial derivative with respect to y:
∂f/∂y = 0 - 12y^2  (since the derivative of 3x with respect to y is 0, and the derivative of 4y with respect to y is 12y^2)

Now, we can substitute x=3 and y=1 into this expression:
∂f/∂y(3,1) = 0 - 12(1)^2 = -12

So, the partial derivative of f(x,y) with respect to y at the point (3,1) is -12.

Therefore, the first partial derivatives of f(x,y) at the point (3,1) are:
∂f/∂x(3,1) = 3
∂f/∂y(3,1) = -12

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The Taylor polynomial t3(x) for the function f(x) = xe−7x centered at a is:

t3(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)2/2! + f'''(a)(x-a)3/3!

To find the Taylor polynomial t3(x), we need to compute the first three derivatives of f(x):

f(x) = xe−7x

f'(x) = e−7x − 7xe−7x

f''(x) = 49xe−7x − 14e−7x

f'''(x) = −343xe−7x + 147e−7x

Next, we evaluate these derivatives at x = a and simplify:

f(a) = ae−7a

f'(a) = e−7a − 7ae−7a

f''(a) = 49ae−7a − 14e−7a

f'''(a) = −343ae−7a + 147e−7a

Now, we plug these values into the formula for t3(x):

t3(x) = ae−7a + (e−7a − 7ae−7a)(x-a) + (49ae−7a − 14e−7a)(x-a)2/2! + (−343ae−7a + 147e−7a)(x-a)3/3!

We can simplify this expression to obtain the final form of t3(x):

t3(x) = ae−7a + (x-a)e−7a(1-7(x-a)) + (x-a)2e−7a(49a-7) + (x-a)3e−7a(-343a+147)/6

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The  inverse Laplace transform of L{f(t)} is:

f(t) = L^-1{2/s} + L^-1{3/s^2} + L^-1{4} + L^-1{13/s^2}
    = 2 + 3t + 4δ(t) + 13t

Thus, f(t) = 2 + 16t for t > 0, and f(t) = 2 for t = 0.

We are given the Laplace transform of a function f(t) as:

L{f(t)} = 2s/(s^2) + 3/(s^2) + 4s/(s^2) + 13/(s^2)

We can simplify this expression as:

L{f(t)} = 2/s + 3/s^2 + 4 + 13/s^2

To find f(t), we need to take the inverse Laplace transform of each term in this expression. We can use the following formulas:

L{t^n} = n!/s^(n+1)
L{e^at} = 1/(s-a)

Using these formulas, we can find that the inverse Laplace transform of each term is:

L^-1{2/s} = 2
L^-1{3/s^2} = 3t
L^-1{4} = 4δ(t)
L^-1{13/s^2} = 13t

where δ(t) is the Dirac delta function.

Therefore, the inverse Laplace transform of L{f(t)} is:

f(t) = L^-1{2/s} + L^-1{3/s^2} + L^-1{4} + L^-1{13/s^2}
    = 2 + 3t + 4δ(t) + 13t

Thus, f(t) = 2 + 16t for t > 0, and f(t) = 2 for t = 0.

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Answer:

  -1572864

Step-by-step explanation:

You want the 19th term of the geometric sequence with first term -6 and common ratio -2.

The n-th term of a geometric sequence is ...

  an = a1·r^(n-1)

where a1 is the first term, and r is the common ratio.

Using the given values of a1 and r, the 19th term is ...

  a19 = (-6)·(-2)^(19-1) = -1572864

<95141404393>

Answer:

Mandy scored a total of 22 points in the basketball game. She made 9 field points, which can be worth either 2 or 3 points. Let's assume that she made x three-point goals and y two-point goals.Then, we can set up the following system of equations:x + y = 9 (because she made a total of 9 field points)3x + 2y = 22 (because the total point value of her field goals was 22).

Solving this system of equations, we can first multiply the first equation by 2 to get:2x + 2y = 18Then, we can subtract this equation from the second equation to eliminate y:3x + 2y - (2x + 2y) = 22 - 18Simplifying this gives:x = 4

Therefore, Mandy made a total of 4 three-point goals and 5 two-point goals in the game.

The volume of the region in the first octant bounded by the coordinate planes, the plane y z=12, and the cylinder x=144−y2 is 432 cubic units. To find the volume of the region in the first octant bounded by the coordinate planes, the plane y z=12, and the cylinder x=144−y2, we need to set up a triple integral.

Since the region is in the first octant, we have the following limits of integration:
0 ≤ x ≤ 144 - y^2
0 ≤ y ≤ √(12/z)
0 ≤ z ≤ 12
So the volume V of the region is given by the triple integral:
V = ∫∫∫ R dV
Where R is the region defined by the above limits of integration, and dV = dxdydz is the differential volume element. Substituting in the limits of integration, we have:
V = ∫0^12 ∫0^√(12/z) ∫0^(144-y^2) dxdydz
Evaluating the integral using the order dzdydx, we get:
V = ∫0^12 ∫0^√(12/z) (144-y^2)dydz
   = ∫0^12 [144y - (1/3)y^3]0^√(12/z) dz
   = ∫0^12 [144√(12/z) - (1/3)(12/z)^(3/2)]dz
   = 576∫0^1 (1 - u^3)du          (where u = √(12/z))
Evaluating the final integral, we get:
V = 576(1 - 1/4)
 = 432 cubic units.

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The quadratic function f(x) = x² + 10x + 37 from standard form to vertex form is f(x) = (x + 5)² + 12

From the question, we have the following parameters that can be used in our computation:

f(x) = x² + 10x + 37

The above quadratic function is its standard form

f(x) = ax² + bx + c

Start by calculating the axis of symmetry using

h = -b/2a

So, we have

h = -10/2

h = -5

Next, we have

f(-5) = (-5)² + 10(-5) + 37

k = 12

The vertex form is then represented as

f(x) = a(x - h)² + k

So, we have

f(x) = (x + 5)² + 12

Hence, the vertex form is f(x) = (x + 5)² + 12

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If the marked price is $816 and the selling price is $800, the discount offered is $16, which is 1.96 percent off the marked price.

The discount refers to the percentage off the marked price of an item.

The discount amount is the dollar value that is taken off the marked price before arriving at the selling price, also known as the discounted price.

The marked price of the item = $816

The selling price (discounted price) = $800

The discount amount in dollars = $16 ($816 - $800)

The discount percentage = 1.96% ($16/$816 x 100)

Thus, the discount that the retailer offered the customer is $16, which translates to 1.96% off the marked price.

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The value of the definite integral (3x-4)²dx is (3b³ - 12b² + 16b + C) - (3a³ - 12a² + 16a + C), since the limits are not mentioned.

To evaluate the definite integral of (3x-4)² dx, we first need to expand the expression and then find the antiderivative. Finally, we need to apply the limits of integration if they are provided.

Expanding the expression:

(3x-4)² = 9x² - 24x + 16
Finding the antiderivative:

∫(9x² - 24x + 16) dx = 3x³ - 12x² + 16x + C
Now, if we have limits of integration (a, b), we would evaluate the antiderivative at those points and subtract the results:
F(b) - F(a) =[tex](3b^3 - 12b^2[/tex] [tex]+ 16b + C[/tex]) - [tex](3a^3 - 12a^2 + 16a + C)[/tex]
However, since no limits of integration were provided, we cannot evaluate the definite integral further.

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The product of (five to the -1st power) and (five to the -3rd power) can be calculated using the properties of exponents. So, the product of (five to the -1st power) and (five to the -3rd power) is equal to 1/625.

When multiplying two expressions with the same base (in this case, five) and different exponents, you can simply add the exponents together. So, for this problem, you will add the exponents -1 and -3, resulting in an exponent of -4.

Therefore, the product of (five to the -1st power) and (five to the -3rd power) is equal to five to the -4th power. To express this as a positive exponent, you can rewrite it as a fraction with the exponent in the denominator: 1/(five to the 4th power). Now, calculate the value: 1/(5^4) = 1/625.

In conclusion, the product of (five to the -1st power) and (five to the -3rd power) is equal to 1/625.

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The property to be used is vertical angle theorem and the value of x is 20/3.

Given is a figure in which two lines are intersecting at a point, making two angles,

The angles are = 3x and 20°,

We need to determine the value of x and the property involved.

So, according to figure we can say, the property involved is vertical angle theorem.

Therefore,

3x = 20

x = 20/3

Hence the property to be used is vertical angle theorem and the value of x is 20/3.

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Answer:

-35.375

Step-by-step explanation:

(-1.5+9.5)=8

5/8 =0.625

7+11=18

0.4*18=36/5

=7.2

7.2/-0.2=

-283/8=

-35.375

Answer:

576 (square yards)

Step-by-step explanation:

length of slanted height of triangle = √(6² + 8²)

= √100

= 10.

surface area = area of 2 triangle faces + area of 3 lengths

= 2 (1/2 X 12 X 8) + 3 (10 X 16)

= 576 (square yards)

We can conclude that the left-hand side (AU (BoC) - (AUB) A (AUC)) and the right-hand side (∅) have no common elements, which proves the equality AU (BoC) - (AUB) A (AUC) = ∅.

To prove the equality AU (BoC) - (AUB) A (AUC) = ∅, we need to show that the left-hand side is an empty set.

First, let's break down the expression step by step:

AU (BoC) represents the union of A with the intersection of B and C. This implies that any element in A, or in both B and C, will be included.

(AUB) represents the union of A and B, which includes all elements present in either A or B.

(AUC) represents the union of A and C, which includes all elements present in either A or C.

Now, let's analyze the right-hand side:

(AUB) A (AUC) represents the intersection of (AUB) and (AUC), which includes elements that are common to both sets.

To prove the equality, we need to show that the left-hand side and the right-hand side have no common elements, i.e., their intersection is empty.

If an element belongs to the left-hand side (AU (BoC) - (AUB) A (AUC)), it must either belong to A and not belong to (AUB) A (AUC), or it must belong to (BoC) and not belong to (AUB) A (AUC).

However, if an element belongs to (BoC), it implies that it belongs to both B and C. Since it does not belong to (AUB) A (AUC), it means that it cannot belong to either A or B or C. Similarly, if an element belongs to A, it cannot belong to (AUB) A (AUC).

Therefore, we can conclude that the left-hand side (AU (BoC) - (AUB) A (AUC)) and the right-hand side (∅) have no common elements, which proves the equality AU (BoC) - (AUB) A (AUC) = ∅.

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(a) Polar coordinate r for P, with r>0 and ?<θ<3π/2 is r=7cos(π/4-θ). (b) Cartesian coordinates for point P are (x,y)=(-7cos(π/4-θ),-7sin(π/4-θ)). (c) The curve passes through the origin twice when 0<θ<2π.

(a) To find r for P, we plug in θ=16π/14 into r=7(2cosθ) and simplify using the identity cos(π/4-θ)=cos(π/4)cos(θ)+sin(π/4)sin(θ)=√2/2(cos(θ)+sin(θ)) to obtain r=7cos(π/4-θ).

(b) To convert from polar to Cartesian coordinates, we use the formulas x=r cos(θ) and y=r sin(θ) and plug in r=7cos(π/4-θ) to get x=-7cos(π/4-θ) and y=-7sin(π/4-θ).

(c) The curve passes through the origin when r=0, which occurs when θ=π/2 and θ=3π/2. Since 0<θ<2π covers each θ value exactly once, the curve passes through the origin twice.

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The phasor method involves converting the sinusoidal functions into phasors, which are complex numbers representing the amplitude and phase of the sinusoidal function. The phasor for a sinusoidal function y = A sin(ωt + φ) is A e^(iφ), where A is the amplitude and φ is the phase angle.

(a) To find the amplitude of y, we need to add the phasors of y1, y2, and y3. The phasor for y1 is 11 e^(i0) = 11, the phasor for y2 is 21 e^(i30°), and the phasor for y3 is 7.0 e^(-i50°). Therefore, the phasor for y is:

Y = 11 + 21 e^(i30°) + 7.0 e^(-i50°)

To find the amplitude of Y, we can take the magnitude of this phasor:

|Y| = sqrt[(11)^2 + (21)^2 + (7.0)^2] = 24.2

Therefore, the amplitude of y is 24.2.

(b) To find the phase constant of y, we need to find the angle that the phasor Y makes with the positive real axis. We can write the phasor Y in rectangular form:

Y = (11 + 21 cos 30° - 7.0 cos 50°) + (21 sin 30° - 7.0 sin 50°) i

The angle that the phasor Y makes with the positive real axis is:

tan^(-1)[(21 sin 30° - 7.0 sin 50°) / (11 + 21 cos 30° - 7.0 cos 50°)]

Using a calculator, we find that this angle is approximately -6.5°. Therefore, the phase constant of y is -6.5°, or we can say that the phase angle of the phasor Y is -6.5°.

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The three additional points on the line are (9, 4), (12, 3) and (15, 2)

From the question, we have the following parameters that can be used in our computation:

(3, 6) and (6, 5)

From the above, we can see that

As x increases by 3, the value of y decreases by 1

This means that the slope of the line is -1/3

Also, we can use the following transformation rule to generate the other points

(x + 3, y - 1)

When used, we have

(9, 4), (12, 3) and (15, 2)

Hence, the three additional points on the line are (9, 4), (12, 3) and (15, 2)

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Answer:

Hi,so since this is a triangle with a right angle and that equals to 90 degrees

so we have found our second number.

The sum of angles in a triangle is 180 degrees

so that would be 27+90+x=180

i.e 117 +x =180

solve equation i.e 180 -117=63

therefore x=63 degrees

hope this was helpful

Answer:

90+27+63=180

so the answer is:

X= 63

The "transient-solution" for X'' + 8X' + 12X = 15,  X(O) = 2, X'(0) = 2 is X(t) = (-7/8) × [tex]e^{-6t}[/tex] + (13/8) × [tex]e^{-2t}[/tex] + 5/4.

In order to find the transient solution of given second-order system, we solve the homogeneous equation associated with it and then find the particular solution for non-homogeneous term.

The homogeneous equation is obtained by setting the right-hand side (RHS) of the equation to zero:

X'' + 8X' + 12X = 0

The characteristic-equation is obtained by assuming a solution of the form X(t) = [tex]e^{rt}[/tex]:

r² + 8r + 12 = 0

(r + 2)(r + 6) = 0

So, the two roots are : r = -2 and r = -6,

The general solution of homogeneous equation is given by:

[tex]X_{h(t)}[/tex] = C₁ × [tex]e^{-6t}[/tex] + C₂ × [tex]e^{-2t}[/tex]

Now, we find the particular-solution for the non-homogeneous term, which is 15. Since 15 is a constant, we assume a constant solution for [tex]X_{p(t)[/tex]:

[tex]X_{p(t)[/tex] = k

Substituting this into original equation,

We get,

0 + 8 × 0 + 12 × k = 15,

12k = 15

k = 15/12 = 5/4

So, particular solution is [tex]X_{p(t)[/tex] = 5/4.

The "transient-solution" is sum of homogeneous and particular solutions:

X(t) = [tex]X_{h(t)[/tex] + [tex]X_{p(t)[/tex]

X(t) = C₁ × [tex]e^{-6t}[/tex] + C₂ × [tex]e^{-2t}[/tex] + 5/4, and

X'(t) = -6C₁ × [tex]e^{-6t}[/tex] -2C₂ × [tex]e^{-2t}[/tex] ,

To find the values of C₁ and C₂, we use initial-conditions: X(0) = 2 and X'(0) = 2.

X(0) = C₁ × [tex]e^{-6\times 0}[/tex] + C₂ × [tex]e^{-2\times 0}[/tex] + 5/4,
X(0) = C₁ + C₂ + 5/4,

Since X(0) = 2, We have:

C₁ + C₂ + 5/4 = 2      ...Equation(1)

and Since X'(0) = 2, we have:

3C₁ + C₂ = -1     ....Equation(2)

On Solving equation(1) and equation(2),

We get,

C₁ = -7/8  and C₂ = 13/8,

Substituting the values, the transient-solution can be written as :

X(t) = (-7/8) × [tex]e^{-6t}[/tex] + (13/8) × [tex]e^{-2t}[/tex] + 5/4.

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The given question is incomplete, the complete question is

For the following second-order system and initial conditions, find the transient solution: X'' + 8X' + 12X = 15,  X(O) = 2, X'(0) = 2.

Answer 136

Step-by-step explanation:

Answer:

11/6

Step-by-step explanation:

For the reciprocal just flip it

Answer:

z = 56

Step-by-step explanation:

124 and Z form a straight line so they will add to 180

124+z = 180

z = 180-124

z = 56

Answer:

z = 56°

Step-by-step explanation:

We know that vertically opposite angles are equal.

∴ y = 124°

We know that angles in a straight line are added up to 180°.

∴ y + z = 180

124 + z = 180

z = 180 - 124

z = 56°

x = 56° ( vertically opposite angles ⇒ x = z )

The cost of a CD is $5.94 more than the cost of a DVD. Let's assume that the cost of a DVD is "x" dollars, then the cost of a CD is "x+5.94" dollars.

Using this information, we can write the following equations:

5(x+5.94) + 2x = 113.63 (cost of 5 CDs and 2 DVDs)

6(x+5.94) + 2x = ? (cost of 6 CDs and 2 DVDs)

Solving the first equation for "x", we get x = 12.21. Substituting this value in the second equation, we get the cost of 6 CDs and 2 DVDs as $83.64.

Therefore, the cost of 6 CDs and 2 DVDs would be $83.64 at All Bets Are Off Electronics if 5 CDs and 2 DVDs cost $113.63, and CDs cost $5.94 more than DVDs.

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The multiplication of 8x³ by (x² + 5x - 6) using distribution is 8x⁵ + 40x⁴ - 48x³.

To multiply the polynomial 8x³ by the polynomial (x² + 5x - 6) using distribution, we will distribute each term of the first polynomial (8x³) to every term in the second polynomial (x² + 5x - 6).

Here's the step-by-step process:

Distribute 8x³ to each term of (x² + 5x - 6):

8x³ · x² + 8x³ · 5x + 8x³ · (-6)

Multiply each term:

8x³ · x² = 8x³ · x² = 8x⁵

8x³ · 5x = 40x³⁺¹ = 40x⁴

8x³ · (-6) = -48x³

Combine the resulting terms:

8x⁵ + 40x⁴ - 48x³

Therefore, the multiplication of 8x³ by (x² + 5x - 6) using distribution is 8x⁵ + 40x⁴ - 48x³.

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The car travels 1.40 meters in 0.187 seconds before coming to a stop. To answer this question, we need to use the equation of motion: v(t) = v0 + at where v(t) is the velocity at time t, v0 is the initial velocity, a is the acceleration, and t is the time.

(a) To find how much time it takes for the car to stop, we need to find the time when v(t) = 0. Using the given values, we have:

0 = 15 - 80.2t

Solving for t, we get:

t = 15/80.2 = 0.187 seconds

Therefore, it takes the car 0.187 seconds to stop.

(b) To find how far the car travels in this time, we can use the equation:

d(t) = v0t + 0.5at^2

Substituting the given values, we get:

d(t) = 15(0.187) + 0.5(-80.2)(0.187)^2

Simplifying, we get:

d(t) = 1.40 meters

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A certain states might not be visited at all while collecting the Statistics. Statistics for certain states might be visited much less often than others leading to very noisy estimates.

Certain states might not be visited at all while collecting the statistics: In the described manner of estimation, there is a possibility that some states may not be visited during the data collection process. This can result in incomplete or biased estimates if those unvisited states have unique characteristics or play an important role in the overall analysis.

Estimates for certain states might be visited much less often than others leading to very noisy estimates: If the data collection process is not balanced or systematic, certain states may be visited less frequently compared to others. As a result, the estimates for these states could be less reliable and prone to higher levels of uncertainty, leading to noisy or inconsistent results.

Therefore, the correct options are:

Certain states might not be visited at all while collecting the statistics.

Estimates for certain states might be visited much less often than others leading to very noisy estimates.

It is likely that certain states might not be visited at all or may be visited much less frequently than others while collecting statistics, leading to very noisy estimates. This is known as the problem of "sparse data." Therefore, the correct options are:

Certain states might not be visited at all while collecting the statistics.

Statistics for certain states might be visited much less often than others leading to very noisy estimates.

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The expected frequency is 50 hence, the answer is (d) 50.

Expected frequency is the frequency we would expect to see in a particular category or group if the null hypothesis is true. The null hypothesis assumes a specific distribution or pattern in the data, and the expected frequency is calculated based on that assumption.

Here we have

Individuals in a random sample of 150 were asked whether they supported capital punishment.

If the opinions of the individuals are uniformly distributed, then the expected frequency for each group would be the same.

Since there are three groups (yes, no, and no opinion), the expected frequency for each group is:

Expected frequency = Total frequency / Number of groups

= 150/3 = 50

Therefore,

The expected frequency is 50 hence, the answer is (d) 50.

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If the eigenvectors of a are the columns of the identity matrix (i), then a is a diagonal matrix. If the eigenvector matrix p is triangular, then a is a triangular matrix.

If the eigenvectors of a are the columns of the identity matrix (i), then a is a diagonal matrix. This is because the eigenvectors of a diagonal matrix are simply the columns of the identity matrix, and the eigenvectors of a matrix do not change under similarity transformations.

If the eigenvector matrix p is triangular, then a is a triangular matrix. This is because the eigenvector matrix p is related to the matrix a through the equation:

A = PDP⁻¹

where D is a diagonal matrix whose diagonal entries are the eigenvalues of a, and P is the matrix whose columns are the eigenvectors of a. If the matrix P is triangular, then the matrix A is also triangular. This can be seen by noting that the inverse of a triangular matrix is also triangular, and the product of two triangular matrices is also triangular.

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-- The given question is incomplete, the complete question is

"If the eigenvectors of A are the columns of I, then A is what sort of matrix? If the eigenvector matrix P is triangular, what sort of matrix is A?"

Answer:

A. 24 inches

Step-by-step explanation:

A cube is defined by having all three lengths being of equal size,

[tex]V = L^3[/tex]

so to find length from the volume of a cube, you must take the cube root of the volume.

[tex]\sqrt[3]{V} = L[/tex]

[tex]\sqrt[3]{216} = 6[/tex]

That means each side of the cube has a length of 6 inches.

The formula for the perimeter is give below the cube diagram.

[tex]P = 4L[/tex]

Which means we just take the length we found above and multiply it by 4.

[tex]4*6 =24[/tex] inches

Therefore 24 inches is your answer.

The equations of the lines with the given gradients and points are:

1. y = -3x + 16

2. y = -5x + 5

3. y = 4x - 20

To find the equation of a line given its gradient and a point it passes through, we can use the point-slope form of a linear equation:

y - y₁ = m(x - x₁),

where (x₁, y₁) represents the given point and m represents the gradient.

Let's calculate the equations for each given gradient and point:

1. Gradient = -3, Point Q(4,4):

Using the point-slope form:

y - 4 = -3(x - 4)

y - 4 = -3x + 12

y = -3x + 16

2. Gradient = -5, Point P(0,5):

Using the point-slope form:

y - 5 = -5(x - 0)

y - 5 = -5x

y = -5x + 5

3. Gradient = 4, Point A(6,4):

Using the point-slope form:

y - 4 = 4(x - 6)

y - 4 = 4x - 24

y = 4x - 20

Therefore, the equations of the lines with the given gradients and points are:

1. y = -3x + 16

2. y = -5x + 5

3. y = 4x - 20

Learn more about gradients at https://brainly.com/question/24216524

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