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3 years ago

Plzzz help

Mathematics

1 answer:

Zolol [24]3 years ago

6 0

Answer: Each ticket costs $32

=========================================================

Explanation:

Let x be the cost of one ticket. If four people buy a ticket each, then they collectively spend a total of 4x dollars for those tickets. Add on 4*3 = 12 dollars from the fact that they bought the ice cream, and we arrive at the expression 4x+12. This is the total amount spent in dollars for the tickets and ice cream combined.

Set this expression equal to 140 and solve for x

4x+12 = 140

4x = 140-12

4x = 128

x = 128/4

x = 32

This says that each ticket costs $32

If one ticket costs $32, then four of them cost a total of 4*32 = 128 dollars. Add on the $12 spent on the ice cream to get 128+12 = 140, which helps verify we have the correct ticket price.

Another way to verify is to replace x with 32 in the original equation we set up, then simplify like so

4x+12 = 140

4*32+12 = 140

128+12 = 140

140 = 140

This is very similar to the previous verification technique used earlier.

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The cubic ...

f(x) = ax³ +bx² +cx +d

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$x=\dfrac{-2b\pm\sqrt{(2b)^2-4(3a)(c)}}{2(3a)} = \dfrac{-2b\pm\sqrt{4b^2-12ac}}{6a}\\\\x=\dfrac{-b\pm\sqrt{b^2-3ac}}{3a}\qquad\text{extreme point locations when $b^2>3ac$}$

There will be zero or two local extremes. A local extreme cannot occur at the point of inflection, which is where the formula would tell you it is when there is only one.

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Part B tells you the midpoint of the local extremes is x = -b/(3a). (This is half the sum of the x-values of the extreme points.) You will notice these are the same point.

The extreme points are located symmetrically about their midpoint, so are located symmetrically about the point of inflection.

_____

Additional comment

There are other interesting features of cubics with two local extremes. The points where the horizontal tangents meet the graph, together with the point of inflection, have equally-spaced x-coordinates. The point of inflection is the midpoint, both horizontally and vertically, between the local extreme points.

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