Math Problem Statement
Solution
Let's first address the two math problems shown in the image.
Problem 10:
The equation given is:
To solve this:
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Expand the left-hand side:
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Set the equation equal to 18:
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Move all terms to one side to set the equation to 0:
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Factor the quadratic equation:
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Solve for :
So, the solutions to the equation are and .
Problem 11:
The function provided is: where represents the height of the diver at time seconds after launch.
This function is a quadratic equation, and we can analyze it further.
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Maximum Height:
Since the equation is quadratic and has a negative coefficient for , the graph is a parabola that opens downwards. The maximum height occurs at the vertex. The formula to find the time of the vertex is: where and . Substituting the values:
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Height at Maximum: Substitute into the height function to find the maximum height: Simplifying:
Thus, the diver reaches a maximum height of 10.625 metres at 0.25 seconds after launch.
Would you like to explore any part of this further, or do you have specific questions?
Follow-up questions:
- What are the steps to solving any quadratic equation?
- How can the vertex formula help in real-life problems involving parabolas?
- How do you calculate the time it takes for the diver to reach the water?
- How do you interpret the solutions to the quadratic equation in problem 10?
- Can you graph both problems and discuss their geometric interpretations?
Tip: Always check if a quadratic equation can be factored easily before using the quadratic formula.
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Math Problem Analysis
Mathematical Concepts
Quadratic Functions
Parabolas
Vertex of a Parabola
Formulas
Vertex formula t_max = -b/(2a)
Height function h(t) = 5t - 10t^2 + 10
Theorems
Quadratic Equation
Maximum of a Parabola
Suitable Grade Level
Grades 10-12
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