## Problem 2053

Given below is information about a network. Choose one of the following three options: the network is definitely a tree; the network is definitely not a tree; the network may or may not be a tree . Accompany your answer with a brief explanation for your choice.

The network has 21 vertices and bridges.

**Solution:-**

A network is another name for a connected graph. Recall that a connected graph is a graph in which there is a path given from any vertex to any other vertex.

A tree is a network that has no circuits. Trees have three key properties that distinguish them from ordinary network. The first property is the single-path property, which states that in a tree, there is only one path connecting two vertices. The second property is the all-bridges property, which states that in a tree, every edge is a bridge, and that if every edge of a network is a bridge then the network must be a tree. The third property is the N-1 edges property, which states that a tree with N vertices has N – 1 edges, and that a network with N vertices and N – 1 edges must be a tree.

Determine which property is the most relevant for classifying the network.

The all-bridges property is the most relevant because the number of bridges is given.

In order to use the all-bridges property, first use the N – 1 edges property to determine how many edges a network with 21 vertices must have it if is a tree.

If a network has 21 vertices, it must have 21 – 1 = 20 edges if it is a tree.

## Problem 2052

Suppose you borrow 865 for a term of three years at simple interest and 3.19% APR.

Determine the total (principal plus interest ) you must pay back in the loan.

**Solution:-**

The total (principal plus interest) you must pay back on the loan is 947.78.

## Problem 2051

Suppose you borrow 560 for a term of five years at simple interest and 2.06% APR.

Determine the total (principal plus interest) you must pay back on the loan.

**Solution:-**

The future payoff, F, which is principal plus interest, obeys the formula F = P(1 + tr), where P is the principal, r is the APR expressed as a decimal, and t is term of the loan in years.

Since the formula requires that the interest rate be written as a decimal, begin by writing the APR of 2.06% as a decimal. To convert a percentage to a decimal, divide by 100.

2.06% = 0.0206

Identify P. Remember that P is the principal, or present value. It is the amount initially borrowed.

P = 560

Identify t. Remember that is the term of the loan in years.

t = 5

Now substitute 560 for P, 0.0206 for r, and 5 for t in formula.

F = P(1 + tr)

= 560 (1 + 0.0206*5)

617.68

Therefore, the total (principal plus interest ) you must pay back on the loan is 617.68.

## Problem 2050

Arvin’s tuition bill last semester was 6240. If he paid 6552 in tuition this semester, what was the percentage increase on his tuition?

**Solution:-**

Begin by finding the amount by which Arvin’s tuition increased.

6552 – 6240 = 312

His tuition increased by 312 from the previous semester. To express this increase as a percentage, determine what percent of 6240 is 312.

Divide 312 by 6240.

$\frac{312}{6240}$ = 0.05.

Write this decimal as a percentage.

0.05 = 5%

Thus, Arvin’s tuition increased by 5%.

## Problem 2049

A 150-piece puzzle is missing 18% of its pieces from its box. How many pieces are in the box?

**Solution:-**

There are 123 pieces in the box.

## Problem 2048

A 450-piece puzzle missing 6% of its pieces from is box. How many pieces are in the box?

**Solution:-**

If a quantity Q is decreased by x%, the remaining quantity D is given by the following formula.

D=

Use the information in the problem statement to identify Q.

Q = 450

Use the information in the problem statement to identify x.

x = 6

Calculate D.

D =

=

=423

Thus, there are 423 pieces in the box.

## Problem 2047

Complete the sentence below.

Triangles for which two sides and the angle opposite one of them are known (SSA) are referred to as the___________________.

**Solution:-**

Triangles for which two sides and the angle opposite one of them are known (SSA) are referred to as the

__ambiguous case.__

## Problem 2046

Determine if the statement below is true or false.

The law of sine can be used to solve triangles where three sides are known.

**Solution:-**

False.

## Problem 2045

Complete the sentence below.

For a triangle with sides a, b, c, and opposite angles A, B, C, the law of sines states that__________________.

**Solution:-**

For a triangle with sides a, b, c, and opposite angles A, B, C, the law of sines states that

## Problem 2044

Find the equation of a line that passes through (2,14) and is parallel to the graph of y = 3x +4.

Write the equation in slope-intercept form.

**Solution:-**

If the slope and a point on the line are known, we can find the equation of the line in slope-intercept form.

Parallel lines have the same slope. Therefore, the slope of the line that is parallel to the given line, y = 3x +4, is 3.

Find the equation of the line that passes through the point (2,14) with a slope of 3. Substitute 2 for x_{1}, 14 for y_{1}, and 3 for m into point-slope form.

y– y_{1 }= m(x-x_{1})

y – 14 = 3(x – 2)

Solve the equation for y.

y – 14 = 3(x – 2)

y – 14 = 3x – 6

y = 3x + 8

Thus, the equation of the desired line is y = 3x + 8.