A model rocket is launched from a roof into a large field. The path of the rocket can be modeled by the equation y = 0.04x+ 8.3x + 4.3, where x is the horizontal distance, in meters, from the starting point on the roof and y is the height, in meters, of the rocket above the ground. How far horizontally from its starting point will the rocket land?

208.02 m
416.03 m
0.52 m
208.19 m

What method(s) would you choose to solve the equation? Explain your reasoning.

x2 –6x + 8 = 0

1.Square roots; there is no x-term.

2.Factoring; the equation is easily factored.

3.Quadratic formula, completing the square or graphing; the coefficient of x2-term is 1, but the equation cannot be factored.

4.Quadratic formula, graphing; the equation cannot be factored easily since the numbers are large.

How many real-number solutions does the equation have?

7x2 + 8x + 5 = 0 (1 point)

one solution
two solutions
no solutions
infinitely many solutions

To find the horizontal distance at which the rocket will land, we need to find the value of x when y is equal to zero.

Using the equation y = 0.04x + 8.3x + 4.3, we can set y equal to zero:

0 = 0.04x + 8.3x + 4.3

To solve this equation, we can use any of the methods given, such as factoring, quadratic formula, completing the square, or graphing.

In this case, the equation can be factored:

0 = (0.04x + 4.3)(x + 1)

Setting each factor equal to zero:

0.04x + 4.3 = 0 or x + 1 = 0

Solving for x in each equation:

0.04x = -4.3 or x = -1

Dividing both sides of the first equation by 0.04:

x ≈ -107.5 or x = -1

Since we are dealing with distances, the negative solution for x does not make sense. Therefore, the rocket will land at x ≈ -1.52 meters horizontally from its starting point.

Out of the answer choices provided, the closest option is 0.52 m.

To find the horizontal distance from the starting point where the rocket will land, we need to find the x-coordinate of the point where y equals zero.

Given the equation y = 0.04x + 8.3x + 4.3, we can set y equal to zero and solve for x:

0 = 0.04x + 8.3x + 4.3

Combining like terms, we get:

0 = 8.34x + 4.3

To isolate x, we subtract 4.3 from both sides:

-4.3 = 8.34x

Finally, we divide both sides by 8.34:

x = -4.3/8.34

After calculating the value, we find that x is approximately -0.515.

Since we are looking for a horizontal distance, we can take the absolute value, which gives us approximately 0.515 meters.

Therefore, the rocket will land approximately 0.52 meters horizontally from its starting point.

So the correct answer is 0.52 m.

To find the horizontal distance the rocket will land, we need to find the value of x when y = 0.

Given the equation y = 0.04x + 8.3x + 4.3, we set y = 0 and solve for x:

0 = 0.04x + 8.3x + 4.3

To solve this equation, we can combine like terms:

0 = 8.34x + 4.3

Next, we isolate the x-term:

-8.34x = 4.3

Now, we can solve for x by dividing both sides by -8.34:

x = 4.3 / -8.34

Calculating this gives us approximately -0.517, which is approximately -0.52 m (to two decimal places).

Therefore, the rocket will land approximately 0.52 m horizontally from its starting point.

As for the second question:

The equation x^2 – 6x + 8 = 0 can be solved using factoring or the quadratic formula.

If you choose factoring, you can rewrite the equation as (x - 4)(x - 2) = 0. By setting each factor equal to zero, you find that x = 4 or x = 2. Therefore, there are two real-number solutions.

If you choose the quadratic formula, you can plug the coefficients into the formula x = (-b ± √(b^2 - 4ac)) / (2a), where a = 1, b = -6, and c = 8. By substituting these values, you can calculate the solutions x = 4 and x = 2. Hence, it confirms that there are two real-number solutions.

For the last question:

To determine the number of solutions for the equation 7x^2 + 8x + 5 = 0, we can use the discriminant (b^2 - 4ac). If the discriminant is greater than zero, there are two real-number solutions. If the discriminant is equal to zero, there is one real-number solution. If the discriminant is less than zero, there are no real-number solutions.

Let's calculate the discriminant for this equation:

b^2 - 4ac = (8^2) - (4)(7)(5) = 64 - 140 = -76

Since the discriminant is less than zero (-76 < 0), there are no real-number solutions for this equation.

Therefore, the answer is: no solutions.