Bosco

In triangle ABC, the measure of angle A is x degrees, the measure of angle B is 2x degrees and the measure of angle C is 6x degrees. What is the value of x? Express your answer as a decimal to the nearest tenth.    

A = x     B = 2x     C = 6x    

A + B + C  =  180^o    

x + 2x + 6x  =  180^o    

              9x  =  180^o     —>>     x = 20.0^o    

_.    

Rationalize the denominator of 5/(3 - sqrt(3)), and write your answer in the form (A + B*sqrt(3))/C where the fraction is in lowest terms. What is A + B + C?    

5 / (3 – sqrt(3))    

Multiply both top and bottom by (3 + sqrt(3))     

to get the difference of squares in the denom.    

[ 5 / (3 – sqrt(3)) ] • [ (3 + sqrt(3)) / (3 + sqrt(3)) ]    

result  5 • (3 + sqrt(3)) / (9 – 3)    

               15 + 5sqrt(3) / 6    

A = 15     B = 5     C = 6    

A+B+C = 15 + 5 + 6  =  26    

_.    

A city park commission received a donation of playground equipment from a parents' organization. The area of the playground needs to be 256 square yards for the children to use it safely. The playground will be rectangular.

In a different plan, the sides can be of any length as long as the rectangular area remains 256 square yards. What dimensions of the rectangular area provide the least perimeter of fencing?    

You get the most area for the amount of perimeter if the figure is a square.    

sqrt(256) = 16     so the dimensions would be 16 x 16    

_.    

A class of 30 students recently took a test. If 20 students scored 80, 8 students scored 90, and 2 students scored 95, then what was the class average (the mean) on this test?    

                                                       20 x 80  = 1600    

                                                         8 x 90  =   720    

                                                         2 x 95  =   190    

add them up                                                    2510    

                                                    2510 / 30  =  83.67    

_.    

Two liters of a 30%  acid solution are mixed with three liters of a 70% acid solution. What percent of the mixture is acid?    

2 liters @ 30% acid  =  0.6 liter acid & 1.4 liter water    

3 liters @ 70% acid  =  2.1 liter acid & 0.9 liter water    

adding them up            2.7 liter acid & 2.3 liter water    

This is a total 5.0 liter solution, 2.7 liters of which is acid    

2.7 / 5.0  =  0.54  ——>>  The new solution is 54% acid    

Confirm plausibility:  2.7 is a little more than half of 5.0    

_.    

How do you write the following equation in standard form? Determine a, b, and c.    

8x^2+x=-78x2+x=−7    

     8x² + x  =  –78x² + x  =  –7    

     86x² + 7  =  0     That's your standard form.    

     a = 86     b = 0     c = 7    

_.    

Let assume Rohan age is X. If we 1/4 his age and add 8 then we find this is equal to 14. Write this expression in equation?    

The equation is                          (x / 4) + 8 = 14    

Work it on out as follows             x / 4  =  14 – 8    

                                                    x / 4  =  6    

                                                         x  =  24    

_.    

A rectangle in the coordinate plane is shown below.  A line with slope 2 splits this rectangle into two regions with equal area.  What is the y-intercept of this line?  The rectangle has coordinates (-1,-1), (-1,1), (1,1), and (1,-1).    

Those coordinates draw a square, centered on the origin.    

Any straight line through the origin will bisect the square.   

Conversely, any line that bisects the square will pass through the origin.    

Therefore, the y-intercept is  (0, 0)    

_.    

A line and a circle intersect at A and B as shown below. Find the distance between A and B.    
The circle is x^2 + y^2 = 1, and the line is y = x.    

This is an easy one, I don't even have to draw it. 

The circle is centered on the origin with a radius of 1. 

The line crosses through the origin and intersects the circle.  

The distance between A & B is the diameter of the circle.  =  2 units      

_.