Problem 78
Question
The quadratic function $$p(x)=-0.387(x-45)^{2}+2.73(x-45)-3.89$$ gives the percentage (in decimal form) of puffin eggs that hatch during a breeding season in terms of \(x\), the sea surface temperature of the surrounding area, in degrees Fahrenheit. For what temperature is the percentage of hatched puffin eggs a maximum? Find the percentage of hatched eggs at this temperature.
Step-by-Step Solution
Verified Answer
The maximum percentage of hatched puffin eggs occurs at a temperature of \(45\) degrees Fahrenheit. At this temperature, the percentage of hatched eggs is \(-3.89\).
1Step 1: Determine the temperature at which the percentage of hatched puffin eggs is maximum
Since \(p(x)\) has a negative leading coefficient and is in vertex form, it will have a maximum at \(x=45\). So, the temperature that maximizes the percentage of hatched puffin eggs is \(45\) degrees Fahrenheit.
2Step 2: Calculate the maximum percentage of hatched puffin eggs
Substitute \(x=45\) into \(p(x)\) to get the maximum percentage. So, \(p(45)=-0.387(45-45)^{2}+2.73(45-45)-3.89=-3.89\). Hence, the percentage of hatched eggs at this temperature is \(-3.89\).
Key Concepts
vertex form of a quadraticinterpreting quadratic modelsoptimization in quadratic functionsapplications of quadratic functions
vertex form of a quadratic
When dealing with quadratic functions, understanding the vertex form is essential for solving various problems including optimization and interpretation of the graph. The vertex form is written as
\(y=a(x-h)^2+k\),
where \((h, k)\) represents the vertex of the parabola, and the value of 'a' determines the direction and width of the parabola. If 'a' is positive, the parabola opens upwards; if 'a' is negative, as in our puffin egg example, it opens downwards indicating the presence of a maximum point. The term \(x-h\) signifies the horizontal distance from the vertex, while 'k' represents the vertical shift from the origin.
In our original exercise, \(p(x)=-0.387(x-45)^2+2.73(x-45)-3.89\), the quadratic function provides a clear vertex form. Here 'h' is 45, which directly tells us the sea temperature that corresponds to the maximum or minimum percentage of hatched eggs. This makes finding the optimal conditions straightforward without completing the square or using calculus.
\(y=a(x-h)^2+k\),
where \((h, k)\) represents the vertex of the parabola, and the value of 'a' determines the direction and width of the parabola. If 'a' is positive, the parabola opens upwards; if 'a' is negative, as in our puffin egg example, it opens downwards indicating the presence of a maximum point. The term \(x-h\) signifies the horizontal distance from the vertex, while 'k' represents the vertical shift from the origin.
In our original exercise, \(p(x)=-0.387(x-45)^2+2.73(x-45)-3.89\), the quadratic function provides a clear vertex form. Here 'h' is 45, which directly tells us the sea temperature that corresponds to the maximum or minimum percentage of hatched eggs. This makes finding the optimal conditions straightforward without completing the square or using calculus.
interpreting quadratic models
A quadratic model, such as the one presented in our exercise, offers a way to understand complex relationships where the rate of change is not constant. The coefficient 'a' determines the concavity of the parabola, which translates into analyzing the nature of change (increasing or decreasing) and identifying the extreme values (maximums or minimums).
In terms of interpretation, the vertex gives us a valuable point of reference, in our case, indicating the optimal sea surface temperature for puffin eggs to hatch. Moving away from the vertex in either direction along the x-axis (temperature), the percentage of hatched eggs changes according to the quadratic equation. This can inform biologists or environmentalists about the potential impact of climate change on puffin breeding, as they could model expected hatch rates at varying temperatures.
In terms of interpretation, the vertex gives us a valuable point of reference, in our case, indicating the optimal sea surface temperature for puffin eggs to hatch. Moving away from the vertex in either direction along the x-axis (temperature), the percentage of hatched eggs changes according to the quadratic equation. This can inform biologists or environmentalists about the potential impact of climate change on puffin breeding, as they could model expected hatch rates at varying temperatures.
optimization in quadratic functions
Optimization in quadratic functions involves finding the maximum or minimum value of the function, which corresponds to the y-coordinate of the vertex for downwards- or upwards-opening parabolas, respectively. Our puffin egg example showcases a real-world application of optimization, with the negative coefficient indicating a downward-opening parabola and hence a maximum value.
Setting the derivative equal to zero or identifying the vertex allows us to find the optimal value without extensive calculations. For the quadratic function given in the exercise, the temperature for maximum hatch rate is 45°F, right at the vertex. No calculus is needed since the vertex form directly reveals the optimal sea surface temperature.
Setting the derivative equal to zero or identifying the vertex allows us to find the optimal value without extensive calculations. For the quadratic function given in the exercise, the temperature for maximum hatch rate is 45°F, right at the vertex. No calculus is needed since the vertex form directly reveals the optimal sea surface temperature.
applications of quadratic functions
Quadratic functions are not just theoretical constructs but have practical applications in various fields such as physics, finance, engineering, and biology, as seen with the puffin eggs' hatching rates. Whenever a situation involves a peak or a trough - be it a thrown ball's highest point or the most economically efficient level of production - quadratic functions often come into play.
Real-life data that exhibit a rising and then falling trend can be modeled with quadratic equations to predict outcomes, analyze trends, and make informed decisions. In environmental science, quadratic models can be particularly powerful, allowing us to predict biological responses to environmental variables such as temperature, which can be crucial for conservation efforts and policy-making.
Real-life data that exhibit a rising and then falling trend can be modeled with quadratic equations to predict outcomes, analyze trends, and make informed decisions. In environmental science, quadratic models can be particularly powerful, allowing us to predict biological responses to environmental variables such as temperature, which can be crucial for conservation efforts and policy-making.
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