Bosco

Find the largest value of n such that 5x^2+nx+40 can be factored as the product of two linear factors with integer coefficients.     

I think the factors would be     (5x + 1)(x + 40)     

                        multiplies to      5x² + 201x + 40     

                                        so      n = 201     

_.     

    The perimeter of a rectangle is 24 inches. What is the number of inches in the maximum diagonal for this rectangle?     

    It will have the longest diagonal when you scrunch it down to a long, narrow figure.

    Assuming integer lengths, the dimensions are 11 x 1 of this long, narrow rectangle.

    Diagonal  =  sqrt(11² + 1²)     

                    =  sqrt(122)     

                    =  11.045     

_.     

There are 6 and 2 over 3 pounds of walnuts in a container, which will be divided equally into containers that hold 1 and 1 over 2 pounds. This would fill ____ and 4 over 9 containers.    

6 ²/₃  divided by  1 ¹/₂      —>>    ²⁰/₃  divided by  ³/₂     

invert divisor and multiply            ²⁰/₃  times  ²/₃   =   ⁴⁰/₉     

                       reduces to                                      =   4 ⁴/₉     

_.     

Noah was asked to compute the product of two numbers.     

He misread the problem and entered the sum of the two numbers instead.     

Luckily he still got the right answer. One of the two numbers is $8$. What is the other number?      

                                                         8 + x  =  8 • x     

divide both sides by x to obtain     8/x + 1  =  8     

                                                             8/x  =  7     

                                                              7x  =  8     

                                                                x  =  8/7     

check answer     

8 + 8/7  =  64/7     

8 •  8/7  =  64/7     

1.  Nine points are arranged in a 3 x 3 grid, as shown below.     

     In how many different ways can you choose three points, so that they form a triangle?     

     76     

     Let me tell you how I arrived at that number.     

     I postulated that any combination of three dots in that 3x3 grid will form a triangle, except     

     the eight combinations that lie in a straight line; namely, 3 vertical, 3 horizontal, 2 diagonal.     

     I calculated that ₉C₃ = 84 and then subtracted the 8 straight lines that would be formed.     

     I posted this before, but made a careless mistake.  I said  84 – 8 = 72   Sheesh.     

2.  A right isosceles triangle has a hypotenuse of 1.  What is the area of the triangle?     

     As I always say, sketch it.  Call the legs a and b and call the hypotenuse c.     

     By Pythagoras' Theorem, a² + b² = c²     

     a² + b² = c²     since a=b and c=1 then     2a² = 1     

                                                                         a² = 1/2     

                                                                         a¹ = sqrt(1/2)  and b also = sqrt(1/2)     

     Area  =  (1/2) • base • height     

               =  (1/2) • sqrt(1/2) • sqrt(1/2)     

               =  (1/2) • (1/2)     ——>>                   Area = 1/4     

_.     

Sarah bought a binder for $7.25 and six folders that totaled $9.05.     

How much did each folder cost?     

                                                B + 6F  =  9.05     

                                                B           =  7.25     

subtract to obtain                           6F  =  1.80     

                                                         F  =  0.30     

_.     

  In triangle ABC, we have angle A = 90 degrees and sin B = 4/7. Find sin C.     

  Sketch it.   AC = 4     

                    BC = 7     

                    AB = sqrt(7² – 4²) = sqrt(33)     

  sin(C) = AB / BC     

  sin(C) = sqrt(33) / 7     

_.     

Write down the quadratic equation whose roots are x = -7 and $x = 11$ and the coefficient of x^2 is 1.     
Enter your answer in the form "x^2 + bx + c = 0".     

(x + 7)(x – 11)  =  x² – 4x – 77     

_.     

 A deep-sea exploring ship is 250 feet below sea level.     

 It rises at a rate of 5 meters per minute.     

 How far below sea level is the ship after 2 minutes?      

 1 meter = 3.28084 feet     —>>     5 meters = 16.4042 feet     

 250 ft – (16.4042 ft / min • 2 min)  =     

 250 ft –   32.8084 ft                        =  217.1916 ft     

_.     

Among all rectangles that have a perimeter of 84, find the dimensions of the one whose area is largest. Write your answers as fractions reduced to lowest terms.     

The rectangle that has the highest ratio of area to perimeter is a square.     

Therefore, when its perimeter is 84, the sides of that square are 21x21.     

_.