You are a world-class math educator and animator, in the style of 3Blue1Brown. Your goal is to create a high-quality, elegant, and intuitive 90-second animated video that visually explains the solution to the following SAT math problem. The primary focus is not just on getting the answer, but on building deep visual intuition about the geometric properties involved.
[Target Audience]
A high school student preparing for the SAT. They may know the formulas but often struggle to connect them to the visual geometry. The tone should be clear, encouraging, and confidence-building. [Narration Style]
Tone: Calm, clear, and methodical. Use "we" to create a collaborative feeling.
Key Phrases: Use phrases like "Now, let's see what this equation is telling us...", "And here is the key insight...", "Notice something beautiful here...", "So, what does this mean for our line?".
Pacing: Allow moments of silence for the visuals to sink in. The narration should complement the animation, not rush it.---Mathematical Formula 1:
f(x) = 5(0.92)^3x
Question Stem:
The function f is defined by the given equation. The equation can be rewritten as
f(x) = 5(1 - p/100)^x , where p is a constant. Which of the following is closest to
the value of p?
Options:
A) 8
B) 12
C) 22
D) 24
视频信息
答案文本
视频字幕
Let's start by understanding what we're looking at. We have two different ways to write the same exponential function. The first form shows f of x equals 5 times 0.92 to the power of 3x. The second form shows f of x equals 5 times the quantity 1 minus p over 100, all raised to the power of x. Since these represent the same function, their graphs must be identical. Notice that both functions start at the point (0, 5) when x equals zero.
Now, let's see what this equation is telling us. The key insight is that we need to rewrite the first function in standard exponential form. We have f of x equals 5 times 0.92 to the power of 3x. Using the exponent rule that a to the m times n equals a to the m raised to the n, we can rewrite this as 5 times the quantity 0.92 cubed, all raised to the power of x. Let's calculate 0.92 cubed: that's 0.92 times 0.92 times 0.92, which gives us approximately 0.7787. So our function is now f of x equals 5 times 0.7787 to the power of x.
And here is the key insight: since both forms represent the exact same function, their bases must be equal. We have 5 times 0.7787 to the power of x equals 5 times the quantity 1 minus p over 100, all to the power of x. For these to be identical, the bases must match. So we can set up the equation: 0.7787 equals 1 minus p over 100. Notice something beautiful here - we've transformed a complex exponential comparison into a simple algebraic equation.
Now let's solve for p step by step. We start with our equation: 0.7787 equals 1 minus p over 100. To isolate p over 100, we subtract 0.7787 from both sides of 1. This gives us p over 100 equals 1 minus 0.7787, which equals 0.2213. Finally, to find p, we multiply both sides by 100. So p equals 0.2213 times 100, which gives us p equals 22.13. We're almost there!
So, what does this mean for our answer? We calculated p equals 22.13, and now we need to find which option is closest. Let's compare: option A is 8, which differs by 14.13. Option B is 12, differing by 10.13. Option D is 24, differing by 1.87. But option C is 22, which differs by only 0.13. Therefore, the answer is C, 22. This beautiful problem shows how exponential functions can be analyzed by comparing their bases, turning a complex-looking problem into straightforward algebra.