hectictar

The solution is the point that lies on both lines....that is the point where the two lines touch.

It looks like that is the point  (-2, 2)   

x  =  2y - 5

5x - 6y  =  7        Since  x = 2y - 5 , we can substitute  2y - 5  in for  x .

5(2y - 5) - 6y  =  7           Distribute  5  to the terms in partenthesees.

10y - 25 - 6y  =  7            Add  25  to both sides of the equation.

10y - 6y  =  7 + 25           Combine like terms.

4y  =  32        Divide both sides by  4 .

y  =  8            Plug this value for  y  into the first equation to find  x .

x  =  2(8) - 5

x  =  16 - 5

x  =  11          

Let's call the length of the rectangle  L, and the width  W .

perimeter  =  L + L + W + W

perimeter  =  2L  +  2W

100  =  2L  +  2W            Solve this for  L .

100 - 2w  =  2L        Divide through by  2 .

50 - W  =  L

area  =  L * W           Substitute  50 - W  in for  L .

area  =  (50 - W) * W

area  =  50W - W²

area  =  -W² + 50W

The maximum area will be when  W = - 50 / [2(-1)]  =  25

and L  =  50 - W  =  50 - 25  =  25

Ahh....glad you caught that CPhill !!! 

∠GCE  =  ∠CBJ      because they are corresponding angles.

And

∠CBJ  and  ∠DBF  are vertical angles, so  ∠CBJ  =  ∠DBF

∠DBF  =  37º

So...     ∠GCE  =  ∠CBJ  =  ∠DBF  =  37º

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∠ABP  and  ∠CBD  are vertical angles, so  ∠ABP  =  ∠CBD

From the information given, ∠CBD  =  ∠CDB

∠CDB  and  ∠EDQ  are vertical angles, so  ∠CDB  =  ∠EDQ

So....     ∠ABP  =  ∠CBD  =  ∠CDB  =  ∠EDQ

∠ABP  =  ∠EDQ

5x - 7  =  2x + 11

3x  =  18

x  =  6

So we know...      ∠ABP  =  5x - 7  =  5(6) - 7  =  23°

∠ABD  and  ∠ABP  form a straight line, so  ∠ABD + ∠ABP  =  180°

∠ABD + 23°  =  180°

∠ABD  =  180° - 23°  =  157°

Let's call the height of the cone  " h " and the area of its base  " a ".

volume of cone  =  (1/3)(a)(h)

The water will also be in the shape of a cone, with a height that is 3/4 that of the container.

volume of water  =  (1/3)(a)(3/4 * h)

volume of water  =  (3/4) * (1/3)(a)(h)

volume of water  =  (3/4) * volume of cone

volume of water  =  75% of the volume of cone

75% of the cone's volume is filled with water.

I count  4  

When a quadratic equation is in the form  f(x)  =  a(x - h)² + k ,  the vertex is  (h, k)

For this problem, they tell us the vertex is  (-1, 5) , so we know that

f(x)  =  a(x - -1)² + 5

f(x)  =  a(x + 1)² + 5

We know it goes through the point  (2, -13) , which means  f(2) = -13

f(2)  =  a(2 + 1)² + 5       And  f(2) = -13

-13  =  a(2 + 1)² + 5

-13  =  9a + 5

-18  =  9a

-2  =  a

Now we've found  a  , so we know that

Since this is a right triangle, we can use the Pythagorean theorem to find the missing side.

Let's call the missing side length  " x " .

6² + x²  =  12.3²        Subtract  6²  from both sides.

x²  =  12.3² - 6²         Simplify

x²  =  151.29 - 36

x²  =  115.29             Take the positive square root of both sides.

x  =  √115.29   ≈   10.737

But there is a small disagreement here... the side opposite the 30° must be the side with length  6 .

However,  6  is not exactly half of 12.3 . In a true 30-60-90 triangle, the side opposite the 30° angle is half the hypotenuse.

So this can't be exactly a 30-60-90 triangle, but it is close.