Archive for the ‘General history’ Category

Problem 1193

If FGHI is a  rectangle and angles G is equal to 2x + 5, find x.

 

Solution:-

 

2x  + 5 = 90

2x  = 85

X = 42.5

 

Problem 1097

 

What is the formula for the circumference C of a circle of radius r? What is the formula for area A of a circle of radius r?

 

Solution:-

 

The formula for the circumference C of a circle of radius r is C = 2πr.

The formula for the area A of a circle of radius r is A = πr2.

 

 

Problem 1009

Let p, q, and r represent the following simple statements:

p: I lie on the sofa.

q: I take a nap.

r: I go jogging.

Write the following compound statement in its symbolic form.

I lie on the sofa and I take a nap, or I go jogging.

Solution:-

First, notice that this statement contains more than one connective.

I lie on the sofa and I take a nap, or I go jogging.

When compound statement containing more than one connective are expressed in words, commas are used to indicate which simple statement are to be grouped together.

Two simple statement that appear on the same side of a comma are grouped together in parentheses when the statement is written symbolically.

Since the compound statement.

I lie on the sofa and I take a nap appears to the left of the comma, the symbolic statement representing it is grouped within parentheses.

The symbolic statement expressing

I lie on the sofa and I take a nap

Is (p  Λ  q).

Next , write the symbol that represents the connective. Then symbolically write the statement the follows the connective.

I lie on the sofa and I take a nap, or I go jogging.

(p Λ q) V r

The symbolic form of the compound statement

I lie on the sofa and I take a nap, or I go jogging.

Is (p Λ q) V r.

Problem 964

Given the annual interest rate and the compounding period, find I, the interest rate per compounding period.

3.4% compounded quarterly.

 

Solution:-

 

The interest rate per compounding period is i =\frac{r}{m} where r is the annual nominal rate and m is the number of compounding periods per year.

The interest is compounded quarterly, so the compounding period is one quarter. There are 4 quarters in one  year.

m = 4

Divide the annual rate by the number of periods per year.

i= \frac{3.4}{4} = 0.85%

The interest rate per compounding period is 0.85%.

 

 

Problem 743

In a family with 8 children, excluding multiple births, what is the probability of having 8 girls?

Assume that a girl is as likely as a boy at each birth.

 

Solution:-

 

Let the sample space S be of all possible permutations of girls and boys for 8 children.

There are 2 possibilities for each of 8 children. Thus using the multiplication principle, the total number of permutation will be 28. Therefore, the number of element is S is

n(S) = 28 = 256

Let  the event E be the set of all element that correspond to the outcome “having  8 girls”.

Since there is only one way that 8 girls out of 8 children can occur, there is only 1 element in the E.

n(E) = 1

Thus, the probability of having 8 girls is

P(E) = \frac{n(E)}{n(S)}

= \frac{1}{25}

 

Problem 735

Write the arithmetic sequence that has three arithmetic means between -16 and 16.

 

Solution:-

 

an  and a1 are given. Find the common difference by using the explicit formula for arithmetic sequences: an = a1 + (n-1)d.

-16, -8, 0 , 8, 16.

 

Problem 703

The letters x and y represent rectangular coordinates. Write the following equation using polar coordinates (rθ).

x^{2}+y^{2}=1

 

Solution:-

 

x = r * cosθ

y = r * sinθ

give the equation x^2 +y^2 = 1

put the x and y value

rcos^{2}\theta + r sin^{2}\theta =1

r^{2}(cos^{2}\theta +   sin^{2}\theta) =1

r^{2}= 1

Problem 557

 

What is the range of 2x2 + 1.

 

Solution:-

 

The graph of the function is a parabola. The positive 2 in the equation will make the graph open up. The positive one shifts the graph up vertically. The range relates to they-values with respect to the graph. Therefore, the range is  y ≥ 1.

 

Problem 556

 

What is the range of the following function? 

f(x) = (4,6), (5,7),(6,8),(7,9)

 

Solution:-

 

Given a set of ordered pairs, the range is found by identifying the y-coordinates from the set.

 

 

So the range is {6, 7, 8, 9}.

Problem 480

What is the equation of the following graph?

480

Solution:-

Circle with a center at (3, -7) and a radius of 3.

 

(x – 3)2 + (y + 7)2 = 9