Archive for the ‘Algebra’ Category

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)

\approx 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= (1-\frac{x}{100})Q

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 = (1-\frac{x}{100})Q

=(1-\frac{6}{100})450

=423

Thus, there are 423 pieces in the box.

 

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 x1, 14 for y1, and 3 for m into point-slope form.

y– y1 = m(x-x1)

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.

 

Problem 2036

Explain how a graph fails the vertical line test.

 

Solution:-

 

The vertical line test is a procedure to check to see if a graph of a relation is a function. Any vertical line drawn through the graph cannot intersect the graph more than once. If any vertical line intersects more than once, the graph fails the vertical line test, and the graph is a relation. If any vertical line only intersects once, the graph passes the vertical line test, and the graph is a function.

 

Problem 2035

Explain why vertical lines have undefined slope.

 

Solution:-

 

When calculating the slope of a line, you find the change in y-coordinates divided by the change in x-coordinates. For a vertical line, the x-coordinates are constant with no change. Therefore, when you subtract their values, the difference is zero. In a fraction any number divided by zero is considered undefined. That means the slope of a vertical line is undefined.

 

Problem 2034

What is the y-intercept of the line y = -3x = 3 ?

 

Solution:-

 

The y-intercept is represented by b in y = mx + b.

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