Bosco

     In the diagram below, we know \tan \theta = \frac{3}{4}. Find the area of the triangle.     

     The triangle is right, and the hypotenuse is 80.     

     Area of a triangle equals half the product of the base times the height.     

     A  =  (1/2)(48)(64)     

     A  =  1536     

     Recognize that 5 is a factor of 80.  And that 3, 4, 5 is a Pythagorean triplet.     

     So, just multiply all the way across by 16 and you get a 48, 64, 80 right triangle.     

     You know that the legs are 48 and 64 because the longest one is the hypotenuse.     

_.     

A G.P has its 3rd term and 7th term as 45 and 3645 respectively. what is the product of its 2nd and 4th     

Call the common ratio "r"     

1st term     

2nd term    45 / r      

3rd term     45                              45     

4th term     45 • r     

5th term     45 • r • r     

6th term     45 • r • r • r     

7th term     45 • r • r • r • r        3645     

8th term     45 • r • r • r • r • r     

Taking what we know, then         45 • r⁴  =  3645     

                                                           r⁴  =  3645 / 45  =  81     

                                                           r¹  =  3     

4th term is  45 • 3  =  135     

2nd term is 45 / 3   =   15             Ha ha, I bit.  It wasn't even necessary to solve for r.     

                                                     No matter what r is, it cancels out in the 2nd and 4th.     

product of 2nd times 4th  =  2025     

_.     

a)  vertex at (1,6)     

b)  opens downward     

c)  symmetry about x=1     

d)  two x-intercepts:  a little short of x= –1     

                                  a little short of x= +3     

e)  y-intercept at y= +4     

_.     

if a parabola has its vertex at (3,-5) and one of its x-intercepts at x=8, what is the other x-intercept?     

Assuming the parabola is symmetrical     

The x-coord of the vertex would be halfway between the x-intercepts     

From vertex x-coord 3 to x-coord 8 is 5 units in the positive direction    

For the other x-intercept, go the same distance

from the vertex x-coord in the other direction     

From 3 go 5 units in the negative direction     

The other x-intercept is at x = – 2     

_.     

Karen paid 1.20 for 6 pencils and 4 erasers at the school store. Trey paid 1.10 for 8 pencils and 2 erasers at the same store.  How much does a pencil cost?  How much does an eraser cost?     

1.20 for 6 pencils and 4 erasers     —>>     6P + 4E  =  1.20     (equation 1)     

1.10 for 8 pencils and 2 erasers     —>>     8P + 2E  =  1.10     (equation 2)     

multiply both sides (eq 2) by 2                  16P  + 4E  =  2.20     (eq 3)     

and subtract (eq 1) from it                          6P  +  4E  =  1.20     (eq 1)     

                                                                 10P             =  1.00     

                                                                                P  =  0.10     Pencils are 10¢ apiece     

substitute 10¢ back into (eq 1)             (6)(0.10) + 4E  =  1.20     

                                                                   0.60  + 4E  =  1.20     

                                                                               4E  =  0.60     

                                                                                 E  =  0.15     Erasers are 15¢ apiece     

_.     

  In five years, Angela will be the age that Khalil is now. In three years, Khalil will be twice as old as Angela. How old is each now?     

  In five years, Angela will be the age that Khalil is now     

  (A + 5)  =  K     

  In three years, Khalil will be twice as old as Angela     

  (K + 3)  =  2 • (A + 3)     

  Substitute (A + 5) in place of K in the second equation.     

  A + 5 + 3  =  2A + 6     

        A + 8  =  2A + 6     

               2  =  A     

         A + 5  =  K     

         2 + 5  =  K     

               7  =  K     

  Angela is 2 and Khalil is 7 at present.     

_.     

The measures of the angles of a triangle are shown in the figure below.  Solve for x.          

The sum of all three angles of a triangle is 180^o.     

The other two angles total is (59^o + 41^o) = 100^o.     

Therefore, x = 180^o – 100^o     

                  x  =  80^o     

_.     

What is KJL?     

Sides KJ and KL are equal (isoceles triangle).     

Since it's isoceles, angles KJL and KLJ are equal.     

The sum of all three angles of a triangle is 180^o.     

So the sum of KJL and KLJ is (180^o – 72^o) = 108^o     

Since KJL and KLJ are equal, each is half of 108^o.     

Therefore angle KJL is 54^o (i.e., half of 108^o).     

_.     

What is the measure of x?     

I'm saying that the triangles are similar, so     

3 : z  =  z : 30      i.e., ratio of short side to hypotenuse of small triangle     

                           equals ratio of short side to hypotenuse of large triangle.     

                           If they are not similar, don't bother reading any further.    

                                    3 / z  =  z / 30     

crossmultiply to get      z² =  90     

by Pythagoras              x²  =  30² – z²     

                                     x²  =  900 – 90     

                                     x   =  sqrt(810)     

This answer looks counterintuitive because the figure is not drawn to scale.     

_.     

     What is the surface area?     

     four sides plus the base

     [4 times (1/2) • 9 • 10)]  plus  (9 • 9)     

     A  =  180 + 81     

     A  =  262 cm²    

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