two boats depart from a port located at (-10, 0) in a coordinate system measured in kilometers, and they…

two boats depart from a port located at (-10, 0) in a coordinate system measured in kilometers, and they travel in a positive x - direction. the first boat follows a path that can be modeled by a quadratic function with a vertex at (0, 5), and the second boat follows a path that can be modeled by a linear function and passes through the point (10, 4). at what point, besides the common starting location of the port, do the paths of the two boats cross?\n(-6, 0.8)\n(-6, 3.2)\n(6, 3.2)\n(6, 0.8)

two boats depart from a port located at (-10, 0) in a coordinate system measured in kilometers, and they travel in a positive x - direction. the first boat follows a path that can be modeled by a quadratic function with a vertex at (0, 5), and the second boat follows a path that can be modeled by a linear function and passes through the point (10, 4). at what point, besides the common starting location of the port, do the paths of the two boats cross?\n(-6, 0.8)\n(-6, 3.2)\n(6, 3.2)\n(6, 0.8)

Answer

Answer:

C. (6, 3.2)

Explanation:

Step1: Find the quadratic - function

The vertex - form of a quadratic function is (y=a(x - h)^2 + k), where ((h,k)) is the vertex. Given the vertex ((h,k)=(0,5)), the quadratic function is (y = ax^2+5). Since the boat starts at ((- 10,0)), substitute (x=-10) and (y = 0) into the equation: (0=a(-10)^2+5), (100a=-5), (a=-\frac{1}{20}). So the quadratic function is (y =-\frac{1}{20}x^2 + 5).

Step2: Find the linear - function

The slope - intercept form of a linear function is (y=mx + b). The line passes through ((-10,0)) and ((10,4)). First, find the slope (m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{4 - 0}{10-(-10)}=\frac{4}{20}=\frac{1}{5}). Using the point - slope form (y - y_1=m(x - x_1)) with the point ((-10,0)), we have (y-0=\frac{1}{5}(x + 10)), which simplifies to (y=\frac{1}{5}x+2).

Step3: Set the two functions equal to each other

Set (-\frac{1}{20}x^2 + 5=\frac{1}{5}x+2). Multiply through by 20 to clear the fraction: (-x^2 + 100 = 4x+40). Rearrange to get a quadratic equation: (x^2+4x - 60 = 0).

Step4: Solve the quadratic equation

Factor the quadratic equation (x^2+4x - 60=(x + 10)(x - 6)=0). The solutions are (x=-10) and (x = 6). We ignore (x=-10) since it is the starting point.

Step5: Find the y - value

Substitute (x = 6) into the linear function (y=\frac{1}{5}(6)+2=\frac{6}{5}+2=\frac{6 + 10}{5}=3.2). So the intersection point is ((6,3.2)).