Modeling with combined functions | Functions and their graphs | Algebra II | Khan Academy | Summary and Q&A

TL;DR

Ify is continuously building a tower on a growing tree, with the tree growing 0.1 meters a month and the tower growing 0.2 meters a month.

Key Insights

• 💗 Ify is building a tower on top of a growing tree, with the tree growing 0.1 meters a month and the tower growing 0.2 meters a month.
• 🌴 The tree's height is represented by the function A(m) = 5 + 0.1m, where m is the number of months.
• #️⃣ The tower's height is represented by the function B(m) = 2 + 0.2m, where m is the number of months.
• ❤️‍🩹 The total vertical distance between the ground and the top end of the tower is calculated using the formula C(m) = A(m) + B(m).
• 🪜 After adding the functions A(m) and B(m), the formula for C(m) simplifies to C(m) = 7 + 0.3m.
• 🌴 The total height of the tower and tree after a certain number of months can be calculated by plugging in the value of m into A(m) and B(m) and adding them.
• 😥 The height and growth rates of the tree and tower can be used to predict their heights at different points in time.

Transcript

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Questions & Answers

Q: What is the current height of the tree and tower?

The tree is currently 5 meters tall, and the tower is currently 2 meters tall.

Q: How much does the tree and tower grow each month?

The tree grows by 0.1 meters each month, while the tower grows by 0.2 meters each month.

Q: What is the formula for calculating the tree's height after a certain number of months?

The formula for the tree's height, A(m), is 5 + 0.1m, where m represents the number of months.

Q: What is the formula for calculating the tower's height after a certain number of months?

The formula for the tower's height, B(m), is 2 + 0.2m, where m represents the number of months.

Q: What is the formula for calculating the vertical distance between the ground and the top end of the tower?

The formula for the vertical distance, C(m), is A(m) + B(m), is the sum of the heights of the tree and tower at a given month.

Q: How would you calculate the total height of the tower and tree after 5 months?

Plug in m=5 into the formulas A(m) and B(m) to get the tree's height of 5 + (0.1 * 5) = 5.5 meters and the tower's height of 2 + (0.2 * 5) = 3 meters. So, the total height would be 5.5 + 3 = 8.5 meters.

Q: How would you calculate the vertical distance between the ground and the top end of the tower after 3 months?

Plug in m=3 into the formula C(m) = A(m) + B(m) to get the tree's height of 5 + (0.1 * 3) = 5.3 meters and the tower's height of 2 + (0.2 * 3) = 2.6 meters. So, the vertical distance would be 5.3 + 2.6 = 7.9 meters.

Summary & Key Takeaways

• The tree is currently 5 meters tall and grows by 0.1 meters each month.

• The tower is currently 2 meters tall and grows by 0.2 meters each month.

• The function A(m) represents the tree's height, which is initially 5 meters and grows by 0.1m * m, while function B(m) represents the tower's height, initially 2 meters and grows by 0.2m * m.