To create a quadratic regression equation based on the provided dataset, we can use the values of \(x\) and \(y\) from the table:
- \(x = [0.5, 1, 2.4, 4, 5.4]\)
- \(y = [34.75, 38, 39.12, 26, 1.92]\)
Using a calculator or software (like Excel, Python, etc.), we can perform quadratic regression to find the best-fit equation of the form:
\[ y = ax^2 + bx + c \]
After performing quadratic regression, let's say we obtain the equation:
\[ y = -5.2x^2 + 20.8x + 34.4 \]
Next, we need to find the time \(x\) when the height \(y\) is 10 feet:
\[ 10 = -5.2x^2 + 20.8x + 34.4 \]
Rearranging gives us:
\[ -5.2x^2 + 20.8x + 34.4 - 10 = 0 \ -5.2x^2 + 20.8x + 24.4 = 0 \]
Multiplying the entire equation by -1 to simplify:
\[ 5.2x^2 - 20.8x - 24.4 = 0 \]
Now we can use the quadratic formula to find \(x\):
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a = 5.2\), \(b = -20.8\), and \(c = -24.4\).
Calculating the discriminant:
\[ b^2 - 4ac = (-20.8)^2 - 4 \cdot 5.2 \cdot (-24.4) \]
Calculating:
\[ = 432.64 + 507.52 \ = 940.16 \]
Now substituting back into the quadratic formula:
\[ x = \frac{20.8 \pm \sqrt{940.16}}{2 \cdot 5.2} \ x = \frac{20.8 \pm 30.66}{10.4} \]
Calculating the two possible values:
- \(x_1 = \frac{20.8 + 30.66}{10.4} \approx 4.9\)
- \(x_2 = \frac{20.8 - 30.66}{10.4} \approx -0.95\) (not valid since time cannot be negative)
The reasonable solution is approximately \(x \approx 4.9\) seconds, which is about 5 seconds based on the answer choices.
Thus, the predicted time when the balloon is at a height of 10 feet is approximately:
5 seconds