Bob has a recipe for salad dressing. The recipe says to mix 1/4 cup lemon juice, some mustard, and some vinegar to make a total of 5/8 cups salad dressing. How much lemon juice does he need to make a total of 4 cups of salad dressing?

Step-by-step explanation:

1/4*x + 1/8*x + 1/4 * x = 4 cups

5/8 * x = 4 cups                    Multiply both sides by 8/5

5/8 * 8/5 x = 4  8/5

x = 32/5

x = 6.4

That means that each ingredient must be multiplied by 6.4

1/4 * 6.4 = 1.6 cups lemon juice

1/8 * 6.4 = .8  cups mustard

1/4 * 6.4 = 1.6 cups vinegar

Total = 4 cups.

  5.   y=6; de=7; fg=8

  6.   x=4.2; jl=14.2; mn=13

step-by-step explanation:

when chords cross inside a circle, the product of the segment lengths is a constant.

5. dh×eh = fh×gh

  y = gh = dh×eh/fh = 3×4/2 . . . divide by the coefficient of gh

  y = 6

of course, the total chord length is the sum of the lengths of its segments.

  de = dh +eh = 3+4 = 7

  fg = fh +gh = 2+6 = 8

6. this problem works the same way as the previous one.

  x = lk = mk×nk/jk = 6×7/10 = 4.2

  jl = jk +lk = 10 +4.2 = 14.2

  mn = mk +nk = 6 +7 = 13

comment on chords and secants

you can think of the points h (problem 5) and k (problem 6) as points where the chords meet. a chord is part of a secant line, a line that intersects the circle in two points. one can also consider these points (h or k) to be the points where the respective secant lines meet.

then, the product that is a constant is the product of the distance from the meeting point (h or k) to one circle intersection with the distance from that meeting point to the other circle intersection. for problem 5, the constant is the product hd×he = hf×hg.

it turns out that this rule regarding the products of lengths to points of intersection is also true when the secants intersect outside the circle.

that is, you only need to remember one rule to work both kinds of problems.

$32,660

step-by-step explanation: