Answer:
Step-by-step explanation:
<u>Area of trapezoid</u>
Given a trapezoid shape whose parallel sides measure b1 and b2 and whose height (perpendicular to the base) is h, then the area of the trapezoid is given by
The dulcimer has the following dimensions
and height h
Thus
Operating
Is the required polynomial
y = 8x² -4x +4
<h3>Explanation:</h3>I find it quick and easy to let a graphing calculator do quadratic regression on the given points. It gives the coefficients of the equation directly. (See attached.)
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If you want to do this "by hand," you can substitute each of the x and y value pairs into the equation to get three linear equations in a, b, and c. These are generally easy to solve, as the "c" variable can be eliminated right away.
<em>For x = 2</em>
... 28 = a·2² +b·2 +c = 4a +2b +c
<em>For x = 4</em>
... 116 = a·4² +4b +c = 16a +4b +c
<em>For x = -3</em>
... 88 = a·(-3)² +b·(-3) +c = 9a -3b +c
Subtracting the first and third equations from the second, we have ...
... (16a +4b +c) -(4a +2b +c) = (116) -(28) ⇒ 12a +2b = 88
... (16a +4b +c) -(9a -3b +c) = (116) -(88) ⇒ 7a +7b = 28
Dividing the first of these reduced equations by 2, and the second by 7, we have ...
Subtracting the second of these from the first gives ...
... (6a +b) -(a +b) = (44) -(4) ⇒ 5a = 40
From which we find
... a = 8
... b = 4 -a = 4 -8 = -4
We choose the first of the original equations to find c:
... c = 28 -4a -2b = 28 -4·8 -2(-4)
... = 28 -32 +8 = 4
With (a, b, c) = (8, -4, 4), the equation of the parabola is ...
... y = 8x² -4x +4
