()) The value of a limited-edition porcelain doll decreased at first, but then the doll became harder to find, and the value started to increase. The value of the doll in dollars can be modeled by the expression $$ 0.25 t^{2}-3 t+24 $$, where $$ t $$ is the time in years after it was first released. ()) What does the number 24 represent in the expression? the doll's value in dollars when it was first released the doll's value in dollars when it was worth the least the time in years from when the doll was released until its value was the least the time in years from when the doll was released until its value was the same as when it was first released Iubmi

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UpStudy AI · Accepted Answer

The number **24** in the expression $$ 0.25 t^{2} - 3 t + 24 $$ represents the **doll's value in dollars when it was first released**.

Here's why:

- **Quadratic Function Structure:** The given expression is a quadratic function of the form $$ V(t) = at^{2} + bt + c $$, where:
  - $$ a = 0.25 $$ determines the curvature of the parabola.
  - $$ b = -3 $$ affects the position of the vertex.
  - $$ c = 24 $$ is the **y-intercept** of the function.

- **Interpreting $$ t = 0 $$:** When $$ t = 0 $$ (the time when the doll was first released), the value of the doll is:
  $$
  V(0) = 0.25(0)^2 - 3(0) + 24 = 24
  $$
  This means the doll was worth **$24** at the time of its release.

- **Contextual Understanding:** The problem states that the doll's value **decreased at first** and then **increased** as it became harder to find. The quadratic function models this behavior with a minimum point (vertex) representing the lowest value before it starts to rise again.

**Therefore, the correct interpretation of the number 24 is:**

**"The doll's value in dollars when it was first released."**
The number 24 represents the doll's value in dollars when it was first released.

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