Questions: Given the function C(r)=(r-4)(r+7)(r-3) its C-intercept is its r-intercepts are

Given the function C(r)=(r-4)(r+7)(r-3) its C-intercept is its r-intercepts are
Transcript text: Given the function $C(r)=(r-4)(r+7)(r-3)$ its $C$-intercept is $\square$ its $r$-intercepts are $\square$ Question Help: Video Submit Question
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Solution

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Solution Steps

To find the \( C \)-intercept, evaluate the function at \( r = 0 \). For the \( r \)-intercepts, set the function equal to zero and solve for \( r \).

Step 1: Find the \( C \)-intercept

To find the \( C \)-intercept, we evaluate the function \( C(r) \) at \( r = 0 \): \[ C(0) = (0 - 4)(0 + 7)(0 - 3) = (-4)(7)(-3) = 84 \] Thus, the \( C \)-intercept is \( 84 \).

Step 2: Find the \( r \)-intercepts

To find the \( r \)-intercepts, we set the function \( C(r) \) equal to zero: \[ (r - 4)(r + 7)(r - 3) = 0 \] This gives us the solutions: \[ r - 4 = 0 \quad \Rightarrow \quad r = 4 \] \[ r + 7 = 0 \quad \Rightarrow \quad r = -7 \] \[ r - 3 = 0 \quad \Rightarrow \quad r = 3 \] Thus, the \( r \)-intercepts are \( -7, 3, \) and \( 4 \).

Final Answer

The \( C \)-intercept is \( \boxed{84} \) and the \( r \)-intercepts are \( \boxed{-7, 3, 4} \).

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