Let's solve the problem step by step: ### **Step 1: Write the formula for the area of a triangle** The area \(A\) of a triangle with base \(b\) and height \(h\) is: \[ A = \frac{1}{2} b h \] ### **Step 2: Find the rate of change of the area** We need to find \(\frac{dA}{dt}\) given that: - \(\frac{db}{dt} = 3\) in/min (base is **increasing**) - \(\frac{dh}{dt} = -3\) in/min (height is **decreasing**) Differentiate both sides with respect to time \(t\): \[ \frac{dA}{dt} = \frac{1}{2} \left( b \frac{dh}{dt} + h \frac{db}{dt} \right) \] ### **Step 3: Substitute the given rates** \[ \frac{dA}{dt} = \frac{1}{2} \left( b(-3) + h(3) \right) \] \[ \frac{dA}{dt} = \frac{1}{2} \left( 3h - 3b \right) \] \[ \frac{dA}{dt} = \frac{3}{2}(h - b) \] ### **Step 4: Analyze what must be true** - If \(h = b\), then \(\frac{dA}{dt} = 0\): **area is not changing** - If \(h > b\), then \(\frac{dA}{dt} > 0\): **area is increasing** - If \(h < b\), then \(\frac{dA}{dt} < 0\): **area is decreasing** ### **Step 5: What must be true?** The sign of \(\frac{dA}{dt}\) depends on the **current values** of \(h\) and \(b\). There is **no guarantee** that the area must always increase, decrease, or stay the same; it depends on whether \(h\) is greater than, less than, or equal to \(b\). --- ## **Final Answer** > **Nothing can be said about whether the area of the triangle is increasing, decreasing, or staying the same without knowing the current values of the base and height.**