A car magazine wants to determine the maximum rpm of a car. The following graph shows the equation y=−2x2+8x . The equation represents the function for the rpm of the car in thousands, where the rpm is y and the number of seconds is x . The maximum rpm for the car is (1 point) Responses 6,000 rpm 6,000 rpm 8,000 rpm 8,000 rpm 10,000 rpm
In order to determine the maximum rpm of the car, we need to find the vertex of the parabola represented by the equation y = -2x^2 + 8x.
The x-coordinate of the vertex can be found using the formula x = -b/2a, where a = -2 and b = 8 in this case.
x = -8 / (2*(-2))
x = -8 / (-4)
x = 2
To find the y-coordinate of the vertex, we substitute x = 2 back into the equation.
y = -2(2)^2 + 8(2)
y = -2(4) + 16
y = -8 + 16
y = 8
Therefore, the maximum rpm for the car is 8,000 rpm.
The engine returns to idle and the rpm becomes 0 at(1 point) Responses 2 seconds 2 seconds 3 seconds 3 seconds 4 seconds
To find when the rpm becomes 0 (idle), we need to set y = 0 in the equation y = -2x^2 + 8x and solve for x.
0 = -2x^2 + 8x
0 = x(-2x + 8)
0 = x(8 - 2x)
Setting each factor to 0 and solving for x:
x = 0 or 8 - 2x = 0
x = 0 or -2x = -8
x = 0 or x = 4
So, the rpm becomes 0 (idle) at x = 0 (starting point) and x = 4 seconds. The rpm becomes 0 again at 4 seconds when the car engine returns to idle.
The rpm of the car increases for the first(1 point) Responses 2 seconds 2 seconds 3 seconds 3 seconds 4 seconds
To determine when the rpm of the car increases for the first time, we need to look at the first derivative of the rpm function and find where it is positive.
The first derivative of the rpm function y = -2x^2 + 8x is:
y' = -4x + 8
To find when the rpm increases for the first time, we need to find where the derivative is positive. Set y' > 0 and solve for x:
-4x + 8 > 0
-4x > -8
x < 2
This indicates that the rpm of the car increases for the first time at x = 2 seconds.
seconds, then it decreases for the next ____ seconds.
The rpm of the car increases for the first time at 2 seconds, as we found earlier.
To determine how long the rpm decreases for after it starts decreasing, we need to find the point where the rpm function starts decreasing, which is when the derivative changes from positive (increasing) to negative (decreasing).
We already found the derivative to be y' = -4x + 8. To find where the rpm starts decreasing, we need to set y' < 0 and solve for x:
-4x + 8 < 0
-4x < -8
x > 2
This indicates that the rpm of the car starts decreasing after 2 seconds. To find for how long the rpm decreases, we need to find when the rpm reaches its next maximum at the vertex of the parabola, which is 4 seconds (as calculated before).
Therefore, the rpm of the car decreases for 2 seconds after it starts decreasing.
The axis of symmetry is(1 point) Responses 2 seconds 2 seconds 3 seconds 3 seconds 4 seconds
The axis of symmetry of a parabola is given by the equation x = -b / (2a), where the equation of the parabola is y = ax^2 + bx + c.
In this case, the equation of the parabola is y = -2x^2 + 8x.
Comparing this to the standard form ax^2 + bx + c, we have a = -2 and b = 8.
Substitute these values into the formula x = -b / (2a):
x = -8 / (2*(-2))
x = -8 / (-4)
x = 2
Therefore, the axis of symmetry of the parabola represented by the equation y = -2x^2 + 8x is x = 2 seconds.