a small zoo predicts the number of visitors, x(t) given the temperature, t (in degrees celcius), as x(t)=89t-2t^2. To estimate the number of zoo employees needed for x visitors, they use the relation E(x)=0.02x+5

a. Express the number of employees needed in terms of the temperature t.

b. estimate the number of employees needed in the zoo when the temperature reaches 17 degrees Celsius. Round to the nearest interger.

Let's tackle each part of the problem step by step.

### Part (a): Express the number of employees needed in terms of the temperature \( t \).

We are given two functions:
1. The number of visitors as a function of temperature \( t \):
   \[
   x(t) = 89t - 2t^2
   \]
2. The number of employees needed as a function of the number of visitors \( x \):
   \[
   E(x) = 0.02x + 5
   \]

We need to express the number of employees \( E \) in terms of the temperature \( t \).

### Step 1: Substitute \( x(t) \) into \( E(x) \).
Substitute the expression for \( x(t) \) into the equation for \( E(x) \):
\[
E(t) = 0.02(89t - 2t^2) + 5
\]

### Step 2: Simplify the expression.
Distribute the \( 0.02 \) and simplify:
\[
E(t) = 0.02 \times 89t - 0.02 \times 2t^2 + 5
\]
\[
E(t) = 1.78t - 0.04t^2 + 5
\]

So, the number of employees needed in terms of temperature \( t \) is:
\[
E(t) = -0.04t^2 + 1.78t + 5
\]

### Part (b): Estimate the number of employees needed when the temperature reaches 17 degrees Celsius.

To find this, substitute \( t = 17 \) into the expression we found in part (a).

### Step 1: Substitute \( t = 17 \) into \( E(t) \):
\[
E(17) = -0.04(17)^2 + 1.78(17) + 5
\]

### Step 2: Calculate each term.
\[
E(17) = -0.04(289) + 1.78(17) + 5
\]
\[
E(17) = -11.56 + 30.26 + 5
\]
\[
E(17) = 23.7
\]

### Step 3: Round to the nearest integer.
The number of employees needed when the temperature is 17 degrees Celsius is approximately:
\[
E(17) \approx 24 \text{ employees}
\]

Would you like further details or have any questions?

Here are 8 related questions you might find interesting:
1. How would you determine the temperature that maximizes the number of visitors to the zoo?
2. What is the vertex of the quadratic function \( x(t) = 89t - 2t^2 \), and what does it represent?
3. How does the number of employees change as the temperature increases?
4. What temperature would require the minimum number of employees?
5. If the temperature were to reach 25 degrees Celsius, how many employees would be needed?
6. How can you find the rate of change of the number of employees with respect to temperature?
7. What are the practical implications of this model for staffing at the zoo?
8. How sensitive is the model to changes in the coefficients (e.g., 89, 2, 0.02) in the equations?

**Tip:** When working with quadratic functions, finding the vertex can provide useful insights, such as identifying maximum or minimum values relevant to real-world scenarios.

Learn how to estimate the number of zoo employees based on temperature and visitor predictions using quadratic and linear functions. Solve for the number of employees needed when the temperature reaches 17 degrees Celsius. Suitable for high school students studying quadratic equations and linear functions.

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