A fountain in a shopping mall has two parabolic arcs of water intersecting. The equation of one parabola is y = -0.25x^2+2x and the equation of the second parabola is y=-0.205x^2+4x-11.75. How high above the base of the fountain do the parabolas intersect? All dimensions are in feet. Round to the nearest hundredth.
To find the height above the base at which the parabolas intersect, we need to set the two equations equal to each other and solve for x. So, let's equate the two equations:
-0.25x^2 + 2x = -0.205x^2 + 4x - 11.75
Now, let's simplify the equation by combining like terms:
-0.25x^2 + 0.205x^2 + 2x - 4x = -11.75
0.045x^2 - 2x = -11.75
Next, let's move all terms to one side of the equation:
0.045x^2 - 2x + 11.75 = 0
Now, we need to solve this quadratic equation for x. There are different ways to solve quadratic equations, such as factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 0.045, b = -2, and c = 11.75. Substituting these values into the quadratic formula, we get:
x = (-(-2) ± √((-2)^2 - 4(0.045)(11.75))) / (2(0.045))
Simplifying further:
x = (2 ± √(4 - 2.085)) / 0.09
x = (2 ± √1.915) / 0.09
Now, let's calculate the two solutions for x:
x1 = (2 + √1.915) / 0.09 ≈ 21.63
x2 = (2 - √1.915) / 0.09 ≈ -6.30
Since we're looking for the height above the base, we'll only consider the positive value of x, which is approximately 21.63.
Now, let's substitute this value of x into either of the original equations to find the corresponding y-coordinate:
y = -0.25(21.63)^2 + 2(21.63)
Simplifying:
y ≈ 22.94
Therefore, the parabolas intersect at a height of approximately 22.94 feet above the base of the fountain.
To find the height above the base where the two parabolas intersect, we need to find the y-coordinate where the two equations are equal.
Let's set the two equations equal to each other and solve for x:
-0.25x^2 + 2x = -0.205x^2 + 4x - 11.75
First, we can rewrite this equation by moving all terms to one side:
-0.25x^2 + 0.205x^2 + 4x - 2x = 11.75
Combine like terms:
-0.045x^2 + 2x = 11.75
Now, let's simplify this equation further. We can multiply both sides by -1 to make the leading coefficient positive:
0.045x^2 - 2x = -11.75
Next, we bring all terms to one side:
0.045x^2 - 2x + 11.75 = 0
At this point, we have a quadratic equation in standard form: ax^2 + bx + c = 0, where a = 0.045, b = -2, and c = 11.75.
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Substituting the values from our equation into the formula:
x = (-(-2) ± √((-2)^2 - 4(0.045)(11.75))) / (2*0.045)
x = (2 ± √(4 - 4(0.045)(11.75))) / 0.09
x = (2 ± √(4 - 2.235)) / 0.09
x = (2 ± √1.765) / 0.09
x = (2 ± 1.33) / 0.09
This gives us two possible values for x:
x₁ = (2 + 1.33) / 0.09 ≈ 37.44
x₂ = (2 - 1.33) / 0.09 ≈ -14.44
Since we are talking about physical dimensions in a shopping mall fountain, we can reasonably assume that x must be positive. Therefore, we discard the negative value of x.
Now that we have the value of x, we can substitute it into either equation (let's use the first one) to find the corresponding y-coordinate:
y = -0.25(37.44)^2 + 2(37.44)
Calculating this value:
y ≈ -0.25(1402.7536) + 74.88
y ≈ -350.6884 + 74.88
y ≈ -275.8084
Rounding to the nearest hundredth, the height above the base where the two parabolas intersect is approximately -275.81 feet.