sally would like a 90 average on the five math tests this semester. her scores so far are 80, 82, 92, 98. what grade must she earn on her 5th and last test to achieve the 90 average?

Answer:

add up variables and divide by 2

Answer:

b

Step-by-step explanation:

18/38 = 47.36%=47% rounding down

Answer:

option 2

in the first equation y is greater than so it would be shaded above the line and in the second equation y is less than so it would be shaded below the line

Step-by-step explanation:

Answer:

its D

Step-by-step explanation:

Answer:

The answer is D

Step-by-step explanation:

3x+2x = 13-3

5x = 10

x=2

The equation of the plane through the origin and the points (2, –4, 6) and (5, 1, 3) is - 9x + 12y + 11 x = 0.

The general equation of a plane through (0, 0, 0) is a(x - 0) + b(y - 0) + c(z - 0) = 0

Since the plane passes though the origin, the equation of the plane is given by ( x, y, z ) = 0. Simplify it so that you write the equation of the plane in the form a x + b y + c z = 0

ax + by + cx = 0...(1)

It will pass through B(2, -4, 6) and C(5, 1, 3) if

a(2) + b(-4) + c(6) = 0

2a - 4b + 6c = 0

a - 2b + 3c = 0 ...(2)

a(5) + b(1) + c(3) = 0

5a + b + 3c = 0 ... (3)

Solving (2) and (3) by cross-multiplication, we have

\(& \frac{a}{-6-3}=\frac{b}{15-3}=\frac{c}{1+10} \\\)

\(& \Rightarrow \frac{a}{-9}=\frac{b}{12}=\frac{c}{11}=\lambda\) (say) }

\(& \Rightarrow\) a = - 9 \(\lambda\), b = 12 \(\lambda\)  and c = 11 \(\lambda\)

Substituting the values of a, b and c in (1), we get

- 9 \(\lambda\) x + 12 \(\lambda\) y + 11 \(\lambda\) x = 0

- 9x + 12y + 11 x = 0

which is the required equation of the plane.

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Answer:

B. All the other products will not be produced.

Step-by-step explanation:

The answer to what automatically happens to all other products made is (B) All the other products will not be produced.

If a company starts using 100% of its resources on one product, it means that it is devoting all of its available time, money, and labor to that product. This means that there will be no resources left over to produce other products. As a result, all other products will not be produced.

This is because a company's resources are limited. There is only so much time, money, and labor that a company can have. If a company decides to focus all of its resources on one product, it means that it is choosing to give up on producing other products.

This is a common strategy for companies that are trying to launch a new product or that are trying to increase the production of a popular product. By focusing all of their resources on one product, companies can increase their chances of success. However, this strategy also means that other products will not be produced.

Here are some additional things to consider:

The company may be able to produce a limited number of other products, but the quality of those products will likely suffer.

The company may be able to produce other products in the future, but it will need to invest in new resources.

The company may decide to outsource the production of other products to other companies.

The decision of whether or not to focus all of a company's resources on one product is a complex one. There are many factors to consider, such as the company's goals, its resources, and the competitive landscape.

Hence, (B) All the other products will not be produced.

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Answer:

-8r + 6

Step-by-step explanation:

-4r - (4r - 6) [original equation]

-4r - 4r + 6 [distribute the "-"]

-8r + 6 [combine like terms]

Answer:

y = 16

Step-by-step explanation:

From the pattern in the table we can see that x to the power of two results in y (2.5 x 2.5 = 6.25, 9.4 x 9.4 = 88.36, etc.) This means that 4 x 4 = 16, which is the value of y.

Answer:

16

Step-by-step explanation:

Plato/Edmentum

A. The solution to the congruence 3x - 107 ≡ 0 (mod 12) is x ≡ 1 (mod 12).

B. The solution to the congruence 5x + 3 - 102 ≡ 0 (mod 7) is x ≡ 6 (mod 7).

C. The solution to the congruence 66 + 9 ≡ 0 (mod 11) is x ≡ 4 (mod 11).

To solve modulo equations or congruences, we need to find values of x that satisfy the given congruence.

A. For the congruence 3x - 107 ≡ 0 (mod 12), we want to find an x such that when 107 is subtracted from 3x, the result is divisible by 12. Adding 107 to both sides of the congruence, we get 3x ≡ 107 (mod 12). By observing the remainders of 107 when divided by 12, we see that 107 ≡ 11 (mod 12). Therefore, we can rewrite the congruence as 3x ≡ 11 (mod 12). To solve for x, we need to find a number that, when multiplied by 3, gives a remainder of 11 when divided by 12. It turns out that x ≡ 1 (mod 12) satisfies this condition.

B. In the congruence 5x + 3 - 102 ≡ 0 (mod 7), we want to find an x such that when 102 is subtracted from 5x + 3, the result is divisible by 7. Subtracting 3 from both sides of the congruence, we get 5x ≡ 99 (mod 7). Simplifying further, 99 ≡ 1 (mod 7). Hence, the congruence becomes 5x ≡ 1 (mod 7). To find x, we need to find a number that, when multiplied by 5, gives a remainder of 1 when divided by 7. It can be seen that x ≡ 6 (mod 7) satisfies this condition.

C. The congruence 66 + 9 ≡ 0 (mod 11) states that we need to find a value of x for which 66 + 9 is divisible by 11. Evaluating 66 + 9, we find that 66 + 9 ≡ 3 (mod 11). Hence, x ≡ 4 (mod 11) satisfies the given congruence.

Modulo arithmetic or congruences involve working with remainders when dividing numbers. In a congruence of the form a ≡ b (mod m), it means that a and b have the same remainder when divided by m. To solve modulo equations, we manipulate the equation to isolate x and determine the values of x that satisfy the congruence. By observing the patterns in remainders and using properties of modular arithmetic, we can find solutions to these equations.

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Answer:

1. a

2. d

3. none of the above - correct answer is root 133. there has to be a typo I think. if they force you to choose do C.

Step-by-step explanation:

To find the volume of the figure.

volume of the figure=volume of the cone+volume of the rectangle prism.

volume of the figure=72+81=153 inch cube.

By applying the trigonometry ratio, SOH, the angle that the ladder makes with the building is calculated as: 17.5°

Recall:

The diagram attached below depicts the problem given.

∅ = x

Opp = 15 ft

Hyp = 50 ft

sin x = 15/50

x = \(sin^{-1}(15/50)\)

x = 17.5°

In conclusion, by applying the trigonometry ratio, SOH, the angle that the ladder makes with the building is calculated as: 17.5°

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Answer:

10

Step-by-step explanation:

2 butter = 160

160×4=910

Edna should leave at 2:07 PM in order to arrive at Jake's party by 2:30 PM

To determine the time Edna should leave, we need to subtract the travel time from the desired arrival time.

If Edna needs to be at Jake's party at 2:30 PM and it takes her 23 minutes to drive there, she should leave 23 minutes before 2:30 PM.

To calculate the departure time, we subtract 23 minutes from 2:30 PM:

2:30 PM - 23 minutes = 2:07 PM

Therefore, Edna should leave at 2:07 PM in order to arrive at Jake's party by 2:30 PM

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Answer:

C. Rolling a fair cube, with faces labeled one to six, ans the cube landing a number less than six.

Step-by-step explanation:

A. flipping a fair coin, and the coin landing on tails, has only got a 50% chance

B. Picking a red marble has only got 10% chance

C. Rolling the cube and getting a number less than 6 has got 5/6 chance, as, from one to six, 5 numbers are less than six.

D. the spinning an arrow similation has only got a 75% chance.

since 5/6 is greater than 3/4, Simulation C is more likely to occur

The event which is most likely to occur is  Rolling a fair number cube, with faces labeled one to six, and the cube landing a number less than six, correct option is C.

Probability refers to a possibility that deals with the occurrence of random events.

The probability of all the events occurring need to be 1.

The formula of probability is defined as the ratio of a number of favorable outcomes to the total number of outcomes.

P(E) = Number of favorable outcomes / total number of outcomes.

We are given that;

A coin flip, bag of marbles, A fair number cube, an arrow spinner

Now,

The probability of an event is defined as the number of ways the event can occur divided by the total number of possible outcomes.

A. The probability of flipping a fair coin and getting tails is 1/2.

B. The probability of choosing a red marble out of a bag with nine blue marbles and one red marble is 1/10.

C. The probability of rolling a number less than six on a fair number cube is 5/6.

D. The probability of spinning the arrow on a spinner and landing on a number greater than one is 3/4.

Comparing the probabilities, we see that option C has the highest probability, with a probability of 5/6.

Therefore, by probability answer will be rolling a fair number cube and getting a number less than six is the most likely event to occur.

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Answer:

The equation will be x=3l and the value of l is 5 when x=15

Step-by-step explanation:

Proportion is defined as the change in one quantity cause change in other quantity.

Writing x varies directly at l

x∝l

Removing the proportionality symbol introduces a constant of proportionality

\(x=kl\)

x=6 when I=2, Putting values

\(6 = k(2)\\k = \frac{6}{2}\\k = 3\)

The equation will become

\(x = 3l\)

Now to find the value of l when x=15, putting x=15 in equation

\(15 = 3l\\\frac{15}{3} = \frac{3l}{3}\\l =5\)

Hence,

The equation will be x=3l and the value of l is 5 when x=15

Since you can go through the non-zero rows from top to bottom and gradually come to the conclusion that the entry where they are non-zero while all the rows below them are zero forces their coefficient in a linear combination yielding a zero row to be zero, the non-zero rows are linearly independent.

Since there are no non-zero rows in a zero matrix and only one linear combination may generate a zero vector (since there are no coefficients), the limitation to non-zero matrices is unnecessary. Additionally, the empty set is linearly independent.

If you write a collection of vectors as the columns of the matrix A and solve Ax = 0, you can tell if the vectors are linearly independent. If any non-zero solutions exist

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Given statement solution is :- a) Power set for two sets A and B with any 2 elements:

Set A: {1, 2}, Set B: {3, 4}

Power set of A: {{}, {1}, {2}, {1, 2}}

Power set of B: {{}, {3}, {4}, {3, 4}}

b) Power set for two sets A and B with any 3 elements:

Set A: {1, 2, 3}, Set B: {4, 5, 6}

Power set of A: {{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}

Power set of B: {{}, {4}, {5}, {6}, {4, 5}, {4, 6}, {5, 6}, {4, 5, 6}}

c) Power set for two sets A and B with any 4 elements:

Set A: {1, 2, 3, 4}, Set B: {5, 6, 7, 8}

Power set of A: {{}, {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}, {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, {1, 2, 3, 4}}

Power set of B: {{}, {5}, {6}, {7}, {8}, {5, 6}, {5, 7}, {5, 8}, {6, 7}, {6, 8}, {7, 8}, {5, 6, 7}, {5, 6, 8}, {5, 7, 8}, {6, 7, 8},

a) Power set for two sets A and B with any 2 elements:

Set A: {1, 2}, Set B: {3, 4}

Power set of A: {{}, {1}, {2}, {1, 2}}

Power set of B: {{}, {3}, {4}, {3, 4}}

Set A: {apple, banana}, Set B: {cat, dog}

Power set of A: {{}, {apple}, {banana}, {apple, banana}}

Power set of B: {{}, {cat}, {dog}, {cat, dog}}

Set A: {red, blue}, Set B: {circle, square}

Power set of A: {{}, {red}, {blue}, {red, blue}}

Power set of B: {{}, {circle}, {square}, {circle, square}}

b) Power set for two sets A and B with any 3 elements:

Set A: {1, 2, 3}, Set B: {4, 5, 6}

Power set of A: {{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}

Power set of B: {{}, {4}, {5}, {6}, {4, 5}, {4, 6}, {5, 6}, {4, 5, 6}}

Set A: {apple, banana, orange}, Set B: {cat, dog, elephant}

Power set of A: {{}, {apple}, {banana}, {orange}, {apple, banana}, {apple, orange}, {banana, orange}, {apple, banana, orange}}

Power set of B: {{}, {cat}, {dog}, {elephant}, {cat, dog}, {cat, elephant}, {dog, elephant}, {cat, dog, elephant}}

Set A: {red, blue, green}, Set B: {circle, square, triangle}

Power set of A: {{}, {red}, {blue}, {green}, {red, blue}, {red, green}, {blue, green}, {red, blue, green}}

Power set of B: {{}, {circle}, {square}, {triangle}, {circle, square}, {circle, triangle}, {square, triangle}, {circle, square, triangle}}

c) Power set for two sets A and B with any 4 elements:

Set A: {1, 2, 3, 4}, Set B: {5, 6, 7, 8}

Power set of A: {{}, {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}, {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, {1, 2, 3, 4}}

Power set of B: {{}, {5}, {6}, {7}, {8}, {5, 6}, {5, 7}, {5, 8}, {6, 7}, {6, 8}, {7, 8}, {5, 6, 7}, {5, 6, 8}, {5, 7, 8}, {6, 7, 8},

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Answer:

let g be f(x)

g= 4x -3

. 7

make x the subject

7g +3. =x

4

the inverse function of f(x) = 7x+3

4

Answer:

9

Step-by-step explanation:

Answer:

Step-by-step explanation:

22

Answer:

a) Center is (0,0) and radius r = 5

b) Center is (0,0) and radius r = 1

d) Center is (5,3) and radius r = 9

e Center is (-2,1) and radius r = 3

g) Center is (-1,-3) and radius r = 7

h) Center is (0,0) and radius r = 2.5

Step-by-step explanation:

a)

Given circle  x² + y² = 25

The standard form of circle equation is

                   ( x-h)² +(y-0)² = r²

Center is (0,0) and radius r = 5

b)

Given circle  x² + y² = 1

The standard form of circle equation is

                   ( x-h)² +(y-0)² = r²

Center is (0,0) and radius r = 1

d)

Given circle  (x-5)² + (y-3)² = 81

The standard form of circle equation is

                   ( x-h)² +(y-0)² = r²

                   (x-5)² + (y-3)² = 9²

Center is (5,3) and radius r = 9

e)

Given circle  (x+2)² + (y+1)² = 9

The standard form of circle equation is

                   ( x-h)² +(y-0)² = r²

                  (x+2)² + (y+1)² = 3²

Center is (-2,-1) and radius r = 3

g)

Given circle  (x+1)² + (y+3)² = 49

The standard form of circle equation is

                   ( x-h)² +(y-0)² = r²

                  (x+1)² + (y+3)² = 7²

Center is (-1,-3) and radius r = 7

h)

Given circle  x² + y² = 6.25

The standard form of circle equation is

                   ( x-h)² +(y-0)² = r²

   Given circle  x² + y² = (2.5)²

Center is (0,0) and radius r = 2.5

We can use the definitions of trigonometric functions to find the values of various trigonometric expressions.

(a)sin⁡(�+�)=−356

sin(α+β)=−56

3

​

(b)cos⁡(�+�)=−3356

cos(α+β)=−56

33

​

(c)tan⁡(�+�)=311

tan(α+β)=11

3

​

(d)sin⁡(�−�)=−311

sin(α−β)=−11

3

​

(e)cos⁡(�−�)=−3356

cos(α−β)=−56

33

​

(f)tan⁡(�−�)=356

tan(α−β)=56

3

​

We are given that tan⁡(�)=−34

tan(α)=−43

​and

cot⁡(�)=247

cot(β)=7

24

​We can use the definitions of trigonometric functions to find the values of various trigonometric expressions.

(a) To findsin⁡(�+�)

sin(α+β), we can use the formula

sin⁡(�+�)=sin⁡(�)cos⁡(�)+cos⁡(�)sin⁡(�)

sin(α+β)=sin(α)cos(β)+cos(α)sin(β). We need to find the values of

sin⁡(�)

sin(α) and

sin⁡(�)

sin(β).

Given that

tan⁡(�)=−34

tan(α)=−

4

3

​, we can draw a right triangle in the second quadrant with opposite side length 3 and adjacent side length 4. By the Pythagorean theorem, the hypotenuse is 5. Therefore,

sin⁡(�)=−35

sin(α)=

5

−3

​

.

Given that

cot⁡(�)=247

cot(β)=

7

24

​

, we can draw a right triangle in the third quadrant with adjacent side length 24 and opposite side length 7. By the Pythagorean theorem, the hypotenuse is 25. Therefore,

sin⁡(�)=−725

sin(β)=25−7

​.

Now we can substitute the values into the formula:

sin⁡(�+�)=sin⁡(�)cos⁡(�)+cos⁡(�)sin⁡(�)=−35⋅−2425+−45⋅−725=−356

sin(α+β)=sin(α)cos(β)+cos(α)sin(β)=

5

−3

​⋅

25

−24

​+5

−4​

⋅

25

−7

​=−56

3

​

(b) To findcos⁡(�+�)

cos(α+β), we can use the formula

cos⁡(�+�)=cos⁡(�)cos⁡(�)−sin⁡(�)sin⁡(�)

cos(α+β)=cos(α)cos(β)−sin(α)sin(β). We need to find the values of

cos⁡(�)cos(α) andsin⁡(�)sin(β).

Given thattan⁡(�)=−34

tan(α)=−43

​, we can use the right triangle in the second quadrant to find

cos⁡(�)cos(α).

Since cos⁡(�)= adjacent hypotenuse

cos(α)=hypotenuse/adjacent

​, we have

cos⁡(�)=−45

cos(α)=5−4

​

.

Given that

cot⁡(�)=247

cot(β)=

7

24

​

, we can use the right triangle in the third quadrant to find

sin⁡(�)

sin(β). Since

sin⁡(�)=opposite hypotenuse

sin(β)=

hypotenuse

opposite

​

, we have

sin⁡(�)=−725

sin(β)=

25

−7

​

.

Now we can substitute the values into the formula:

cos⁡(�+�)=cos⁡(�)cos⁡(�)−sin⁡(�)sin⁡(�)=−45⋅2425−−35⋅−725=−3356

cos(α+β)=cos(α)cos(β)−sin(α)sin(β)=

5

−4

​

⋅

25

24

​

−

5

−3

​

⋅

25

−7

​

=−

56

33

​

(c) To find

tan⁡(�+�)

tan(α+β), we can use the formula

tan⁡(�+�)=sin⁡(�+�)cos⁡(�+�)

tan(α+β)=

cos(α+β)

sin(α+β)

​

. We have already found

sin⁡(�+�)

sin(α+β) and

cos⁡(�+�)

cos(α+β), so we can substitute the values:

tan⁡(�+�)=sin⁡(�+�)cos⁡(�+�)=−356−3356=311

tan(α+β)=

cos(α+β)

sin(α+β)

​

=

−

56

33

​

−

56

3

​

​

=

11

3

​

(d) To find

sin⁡(�−�)

sin(α−β), we can use the formula

sin⁡(�−�)=sin⁡(�)cos⁡(�)−cos⁡(�)sin⁡(�)

sin(α−β)=sin(α)cos(β)−cos(α)sin(β). We have already found

sin⁡(�)

sin(α),

cos⁡(�)

cos(α),

sin⁡(�)

sin(β), and

cos⁡(�)

cos(β), so we can substitute the values:

sin⁡(�−�)=sin⁡(�)cos⁡(�)−cos⁡(�)sin⁡(�)=−35⋅−2425−−45⋅−725=−311

sin(α−β)=sin(α)cos(β)−cos(α)sin(β)=

5

−3

​

⋅

25

−24

​

−

5

−4

​

⋅

25

−7

​

=−

11

3

​

(e) To find

cos⁡(�−�)

cos(α−β), we can use the formula

cos⁡(�−�)=cos⁡(�)cos⁡(�)+sin⁡(�)sin⁡(�)

cos(α−β)=cos(α)cos(β)+sin(α)sin(β). We have already found

sin⁡(�)

sin(α),

cos⁡(�)

cos(α),

sin⁡(�)

sin(β), and

cos⁡(�)

cos(β), so we can substitute the values:

cos⁡(�−�)=cos⁡(�)cos⁡(�)+sin⁡(�)sin⁡(�)=−45⋅−2425+−35⋅−725=−3356

cos(α−β)=cos(α)cos(β)+sin(α)sin(β)=

5−4​⋅

25−24​+5−3​⋅25−7

​=−5633

​

(f) To find

tan⁡(�−�)

tan(α−β), we can use the formula

tan⁡(�−�)=sin⁡(�−�)cos⁡(�−�)

tan(α−β)=

cos(α−β)

sin(α−β)

​

. We have already found

sin⁡(�−�)

sin(α−β) and

cos⁡(�−�)

cos(α−β), so we can substitute the values:

tan⁡(�−�)=sin⁡(�−�)cos⁡(�−�)=−311−3356=356

tan(α−β)=

cos(α−β)

sin(α−β)

​=−5633−113​

=563

​

(a)sin⁡(�+�)=−356

sin(α+β)=−563

​

(b)cos⁡(�+�)=−3356

cos(α+β)=−5633

​

(c)tan⁡(�+�)=311

tan(α+β)=113

​

(d)sin⁡(�−�)=−311

sin(α−β)=−113

​

(e)cos⁡(�−�)=−3356

cos(α−β)=−5633​(f)tan⁡(�−�)=356

tan(α−β)=563

​

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Answer:

the anwser is a i think im helping you out dont repoty

If the dimensions of trapezoid JKLM are multiplied by a scale factor of f to create trapezoid J'K'L'M' "The area of trapezoid J'K'L'M' is f times the area of trapezoid JKLM." Therefore, option A is correct.

When you multiply the dimensions of a trapezoid by a scale factor, the area of ​​the resulting trapezoid is equal to the square of the scale factor multiplied by the area of ​​the original trapezoid. In this case, the scaling factor is f, so the area of ​​trapezoid J'K'L'M' is f² times the area of ​​trapezoid JKLM.

When scaling a trapezoid, the base perimeter, angle, and length can change, depending on the specific value of the scale factor and the dimensions of the original trapezoid. 

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Answer:

The answer is 85°

Step-by-step explanation:

We know that the sum of all the angles of a triangle equals 180°. Therefore, since we know that <1=45° and <2=50°. If we add this together and then subtract them from 180° we get the missing angle. Visual:

50°+45°=95°

180°-95°=85°

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Answer:

speed of boat is 12 miles per hour and the speed of current is 9 miles per hour

Step-by-step explanation:

A substitute is a good or service that buyers can quickly swap out for another. For instance, a one-dollar bill can be used in place of another dollar bill.

In business and economics, a replacement, or substitutable good, is a good or service that consumers perceive as just being substantially the same as or reasonably similar to some other good. Consumers are given options and alternatives through substitutes, which also spur competition and lower prices in the market.

Here are a few examples of replacement goods:

1. A $1 bill can be exchanged for four quarters.

2. Coke against Pepsi

3. Regular vs. premium gas

4. Butter and lard

5. Tea and coffee

6. Apples and oranges

7. Comparing driving a car to riding a bike

8. Books in general and e-books

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To sketch the curve with the given vector equation \(r(t) = t^2i + t^4j + t^6k,\) we can plot points for various values of t and connect them with a smooth curve that spirals upwards as t increases, and to indicate the direction in which t increases, you can use an arrow pointing in the positive i direction.

Here is a sketch of the curve defined by the vector equation \(r(t) = t^2 i + t^4 j + t^6 k:\)

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To sketch the curve with the given vector equation\(r(t) = t^2i + t^4j + t^6k,\)you can start by plotting points for various values of t and then connecting these points to form a curve.

Choose some values of t, such as t = -1, 0, 1, 2, 3, and 4.

Plug each value of t into the vector equation to get the corresponding vector.

For example, when t = 2, r(2) = 4i + 16j + 64k.

Plot each vector as a point in three-dimensional space.

For example, the vector 4i + 16j + 64k would be plotted at the point (4, 16, 64).

Connect the points with a smooth curve to show the shape of the curve.

To indicate the direction in which t increases, you can use an arrow.

The arrow should point in the direction of increasing t.

In this case, since the coefficient of i (the x-component) is positive and the coefficient of j and k are both positive, the curve will spiral upwards as t increases.

Regarding your substitution, we are correct that \(y = x^2\) and \(z = x^3\) are possible substitutions.

These equations represent a parabolic curve in the xy-plane and a cubic curve in the xz-plane.

However, keep in mind that the vector equation\(r(t) = t^2i + t^4j + t^6k\)already represents a curve in three-dimensional space, so we do not need to make any additional substitutions to sketch the curve.

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To sketch the curve with the given vector equation r(t) = t^2i + t^4j + t^6k, start by identifying the parametric equations: x = t^2, y = t^4, and z = t^6. You already found the possible substitutions, y = x^2 and z = x^3. Now, create a 3D graph with axes for x, y, and z. For various values of t (e.g., -2, -1, 0, 1, 2), calculate the corresponding x, y, and z coordinates using the parametric equations. Plot these points on the graph and connect them to form the curve. Add an arrow to indicate the direction in which t increases, which is the direction of the curve as you move from negative to positive t values.

 To sketch the curve with the given vector equation, you can start by plotting points on the coordinate plane. The vector equation r(t) = t2i + t4j + t6k tells us that the x-coordinate is t^2, the y-coordinate is t^4, and the z-coordinate is t^6. You can choose a few values of t, such as -1, 0, and 1, and plug them into the equation to get the corresponding points on the curve.

To indicate the direction in which t increases, you can use an arrow. Since t is a parameter, it can increase in either the positive or negative direction, depending on the direction in which you choose to move along the curve. You can use an arrow to show the direction of increasing t, which will help you visualize the direction of the curve.

As for the possible substitution, y = x^2 and z = x^3 are indeed possible substitutions, since they satisfy the equations x = t^2, y = t^4, and z = t^6. Substituting these expressions for x, y, and z will give you a simpler representation of the curve, which can make it easier to sketch.

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Answer:

45

Step-by-step explanation:

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