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This table shows the profit for a company (in millions of dollars) in different years.

| Year | Profit (millions [tex]$) |
|------|---------------------|
| 1 | 8 |
| 4 | 15 |
| 6 | 17 |
| 8 | 17 |
| 10 | 15 |
| 12 | 6 |

The quadratic regression equation that models these data is \( y = -0.34x^2 + 4.43x + 3.46 \).

Using the quadratic regression equation, what was the predicted profit in year 4?
A. $[/tex]19.82 million
B. $15.74 million



Answer :

GinnyAnswer
To determine the predicted profit in the year 4 using the given quadratic regression equation [tex]\( y = -0.34x^2 + 4.43x + 3.46 \)[/tex], we need to follow these steps:

1. Identify the value of [tex]\( x \)[/tex]:
- The year given is 4, so [tex]\( x = 4 \)[/tex].

2. Plug the value of [tex]\( x \)[/tex] into the quadratic regression equation:
- The quadratic regression equation given is [tex]\( y = -0.34x^2 + 4.43x + 3.46 \)[/tex].

3. Substitute [tex]\( x = 4 \)[/tex] into the equation:
- Replace [tex]\( x \)[/tex] with 4 in the equation:
[tex]\[ y = -0.34(4)^2 + 4.43(4) + 3.46 \][/tex]

4. Calculate each term separately:
- First term [tex]\( -0.34(4)^2 \)[/tex]:
[tex]\[ 4^2 = 16 \][/tex]
[tex]\[ -0.34 \times 16 = -5.44 \][/tex]

- Second term [tex]\( 4.43(4) \)[/tex]:
[tex]\[ 4.43 \times 4 = 17.72 \][/tex]

- Combine all terms together:
[tex]\[ y = -5.44 + 17.72 + 3.46 \][/tex]

5. Add the results:
- Adding the terms together:
[tex]\[ -5.44 + 17.72 = 12.28 \][/tex]
[tex]\[ 12.28 + 3.46 = 15.74 \][/tex]

6. Determine the predicted profit:
- The predicted profit for the year 4 is [tex]\( \$ 15.74 \)[/tex] million.

Among the given options, the correct prediction of the profit in year 4 is:

B. [tex]$\$[/tex] 15.74$ million

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