Question

an ornithologist studying the flight of birds measured the efficiency for parakeets flying at various speed in a descending flight pattern 0.18 at 12 mph, 0.23 at 22 mph and 0.15 at 30mph. the measurements suggest the efficiency E is a quadratic function of the speed s. find the quadratic function whose graph goes through the three given ordered pairs, and find the speed that gives the maximum efficiency for descending flight.

Answer 1

To find the quadratic function that represents the efficiency E as a function of the speed s and the speed that gives the maximum efficiency for descending flight, we can set up a system of equations using the given ordered pairs, solve for the values of a, b, and c, and find the x-coordinate of the vertex of the quadratic function.

Step-by-step explanation:

To find the quadratic function that represents the efficiency E as a function of the speed s, we need to use the three given ordered pairs (12, 0.18), (22, 0.23), and (30, 0.15).

Let's start by setting up a system of equations:

0.18 = a(12²) + b(12) + c

0.23 = a(22²) + b(22) + c

0.15 = a(30²) + b(30) + c

Solving this system of equations will give us the values of a, b, and c. Once we know those values, we can write the quadratic function in the form of E(s) = as² + bs + c.

E(s) = 0.00106s² - 0.00083s + c

To find the value of c, use any of the three original equations. Let's use equation 1:

0.18 = 0.00106(12)² - 0.00083(12) + c

0.18 = 1.58416 - 0.01 + c

c = 0.27416

Therefore, the quadratic function representing the efficiency of parakeets in descending flight is:

E(s) = 0.00106s² - 0.00083s + 0.27416

Finding the Speed for Maximum Efficiency

To find the speed that gives the maximum efficiency, we need to find the vertex of the parabola represented by the quadratic function. The vertex corresponds to the maximum point of the parabola.

The x-coordinate of the vertex is given by:

s_max = -b / (2a)

Substituting the values of a and b from the function:

s_max = -(-0.00083) / (2 × 0.00106)

s_max ≈ 19.51 mph

Therefore, the parakeets achieve their maximum efficiency for descending flight at a speed of approximately 19.51 mph.

Answer 2

Step-by-step explanation:

Given:

1. when s = 12 mph t =0.18 let (s,t) or (a,b) = (12,0.18)
2. when s = 22 mph t = 0.23 s (s,t) or (a,b) =(22,0.23)
3. when s = 30 mph t = 0.15 s (s,t) or (a,b) =(30,0.15)

we know the general format of Quadratic equation to be

= ax²+bx+c=0

Step 1 :

when the birds pass through a,b (12,0.18) we cane give the expression as

= ax²+bx+c=0

= 12x² + 0.18x + c= 0

it can also be written as

s = ut + 1/2 at² = 0.18t +( 1/2) a(0.18)²

step 2 :

through points (22,0.23)

= 22x² + 0.23x + c = 0

s = ut + 1/2 at² = 0.23t + (1/2) a(0.23)²

Step 3 :

through points (30,0.15)

= 30x² + 0.15x + c = 0

s = ut + 1/2 at²

= 0.15 u +(1/2) a(0.15)²