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alexira [117]
3 years ago
10

The simple formula for the nth term of an arithmetic sequence is an = 5n + 15. What is the explicit formula corresponding to the

simple formula?
A. an = 15 + 5n
B. an = 15 + (n - 1)(5)
C. an = 20 + (n)/(5)
D. an = 20 + (n - 1)(5)
E. an = 5 + (n - 1)(15)
Mathematics
1 answer:
Nikitich [7]3 years ago
7 0
Sequence will look like this:
Term 1: 20 --------- 20 + 5(0)
Term 2: 25 --------- 20 + 5(1)
Term 3: 30 --------- 20 + 5(2)
Term 4: 35 --------- 25 + 5(3)
ETC.

It cab noted that the numbers in the bracket (that is, 0, 1, 2, 3, 3, etc) = n-1

Then, the explicit formula will be;
an = 20 + 5(n-1).

The correct answer is D.
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Mr. Nordman gets a commission of $2.30 on each pair of women's shoes he sells, and a commission of $3 on each pair of men's shoe
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Answer:

x ≥ 5

y ≥ 10

3x + 2.3y ≥ 60

Step-by-step explanation:

Lets the number of men's shoes he sells be x and women's shoes be y.

For each men's shoes , he gets a commission of $3 and for each women's shoe, he gets a commission of $2.30 .

He needs to sell atleast 10 women's shoes and 5 men's shoes.

Also he needs to make atleast $60 per week.

x ≥ 5

y ≥ 10

3x + 2.3y ≥ 60

These are are the 3 required inequalities.

Now we will see an example.

If he sells 20 womens shoes and 10 mens shoes , he will meet the requirements.

He will make a total profit = 2.3\times 20 + 3\times 10 =$76

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Find equations of the tangent plane and the normal line to the given surface at the specified point. x + y + z = 8exyz, (0, 0, 8
Dima020 [189]

Let f(x,y,z)=x+y+z-8e^{xyz}. The tangent plane to the surface at (0, 0, 8) is

\nabla f(0,0,8)\cdot(x,y,z-8)=0

The gradient is

\nabla f(x,y,z)=\left(1-8yze^{xyz},1-8xze^{xyz},1-8xye^{xyz}\right)

so the tangent plane's equation is

(1,1,1)\cdot(x,y,z-8)=0\implies x+y+(z-8)=0\implies x+y+z=8

The normal vector to the plane at (0, 0, 8) is the same as the gradient of the surface at this point, (1, 1, 1). We can get all points along the line containing this vector by scaling the vector by t, then ensure it passes through (0, 0, 8) by translating the line so that it does. Then the line has parametric equation

(1,1,1)t+(0,0,8)=(t,t,t+8)

or x(t)=t, y(t)=t, and z(t)=t+8.

(See the attached plot; the given surface is orange, (0, 0, 8) is the black point, the tangent plane is blue, and the red line is the normal at this point)

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Formulate the situation as a system of two linear equations in two variables. Be sure to state clearly the meaning of your x- an
slega [8]

Answer:

1) There were 33 $4,000 investors and 27 $8,000 investors.

2) The solution in x = 4, y = 9.

3) There were 24 nickels and 56 dimes.

Step-by-step explanation:

1) A lawyer has found 60 investors for a limited partnership to purchase an inner-city apartment building, with each contributing either $4,000 or $8,000. If the partnership raised $348,000, then how many investors contributed $4,000 and how many contributed $8,000?

I am going to say that:

x is the number of investors that contributed 4,000.

y is the number of investors that contributed 8,000.

Building the system:

There are 60 investors. So:

x + y = 60

In all, the partnership raised $348,000. So:

4000x + 8000y = 348000

I am going to simplify by 4000. So:

x + 2y = 87

Solving the system:

The elimination method is a method in which we can transform the system such that one variable can be canceled by addition. So:

1)x + y = 60

2)x + 2y = 87

I am going to multiply 1) by -1. So we have

1)-x - y = -60

2)x + 2y = 87

By addition, the x are going to cancel each other

-x + x - y + 2y = -60 + 87

y = 27

For x:

x + y = 60

x = 60-y = 60-27 = 33

There were 33 $4,000 investors and 27 $8,000 investors.

2) Solve the system by row-reducing the corresponding augmented matrix.

2x + y = 17

x + y = 13

This system has the following augmented matrix:

\left[\begin{array}{ccc}2&1&17\\1&1&13\end{array}\right]

To help the row reducing, i am going to swap the first with the second line:

L1  L2

So we have:

\left[\begin{array}{ccc}1&1&13\\2&1&17\end{array}\right]

Now, reducing the first column.

L2 = L2 - 2L1

So we have:

\left[\begin{array}{ccc}1&1&13\\0&-1&-9\end{array}\right]

Now we do:

L2 = -L2

And the matrix is:

\left[\begin{array}{ccc}1&1&13\\0&1&9\end{array}\right]

Now to reduce the second column, we do:

L1 = L1 - L2

\left[\begin{array}{ccc}1&0&4\\0&1&9\end{array}\right].

So the solution is:

x = 4, y = 9.

3) A jar contains 80 nickels and dimes worth $6.80. How many of each kind of coin are in the jar?

I am going to say that x is the number of nickels and y is the number of dimes.

Each nickel is worth 5 cents and each dime is worth 10 cents.

Building the system:

There are 80 coins in all:

x + y = 80

They are worth $6.80. So:

0.05x + 0.10y = 6.80

Solving the system:

1)x + y = 80

2)0.05x + 0.10y = 6.80

I am going to divide 1) by -10, so we can cancel y. So:

1)-0.10x - 0.10y = -8

2)0.05x + 0.10y = 6.80

Adding:

-0.10x + 0.05x - 0.10y + 0.10y = -8 + 6.80

-0.05x = -1.2 *(-100)

5x = 120

x = \frac{120}{5}

x = 24

Also

x + y = 80

y = 80-x = 80-24 = 56

There were 24 nickels and 56 dimes.

8 0
3 years ago
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