Question

Kayla set up an outdoor digital thermometer to record the temperature overnight as part of her science fair project. She began recording the temperature, in degrees Fahrenheit at 10 pm. Kayla modeled the overnight temperature with function t, where h represents the number of hours since 10pm. t(h) = 0.5 h^2 - 5h + 27.5 What is the lowest temperature and at what time did it occur?

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Eliseo

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1. Identify the given quadratic function:

t(h) = 0.5h^2 - 5h + 27.5

2. To find the minimum temperature, we need to find the vertex of the quadratic function since it opens upwards (the coefficient of \( h^2 \) is positive).

3. The formula for the x-coordinate of the vertex of a parabola \( ax^2 + bx + c \) is:

h = -\frac{b}{2a}

4. Given \( a = 0.5 \) and \( b = -5 \):

h = -\frac{-5}{2 \cdot 0.5}

h = \frac{5}{1}

h = 5

5. Substitute \( h = 5 \) back into the function to find the temperature at this time:

t(5) = 0.5(5)^2 - 5(5) + 27.5

t(5) = 0.5 \cdot 25 - 25 + 27.5

t(5) = 12.5 - 25 + 27.5

t(5) = 15

6. Therefore, the lowest temperature occurs at \( h = 5 \) and the temperature is:

t_{\text{min}} = 15 \; \text{degrees Fahrenheit}

7. Since \( h \) represents the number of hours since 10 pm, the lowest temperature occurred at:

10 \; \text{pm} + 5 \; \text{hours} = 3 \; \text{am}

Final answer:

t_{\text{min}} = 15 \; \text{degrees Fahrenheit at 3 am}

2. To find the minimum temperature, we need to find the vertex of the quadratic function since it opens upwards (the coefficient of \( h^2 \) is positive).

3. The formula for the x-coordinate of the vertex of a parabola \( ax^2 + bx + c \) is:

4. Given \( a = 0.5 \) and \( b = -5 \):

5. Substitute \( h = 5 \) back into the function to find the temperature at this time:

6. Therefore, the lowest temperature occurs at \( h = 5 \) and the temperature is:

7. Since \( h \) represents the number of hours since 10 pm, the lowest temperature occurred at:

Final answer:

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