The deflection equation of a cantilever beam is given by \[ E \mathrm{lv}=93.75 \mathrm{x}^{3}-30<\mathrm{x}-3>^{3}-5<\mathrm{x}-6>^{4}-298 \mathrm{x}\left(\mathrm{kNm} \mathrm{~m}^{3}\right) \] where \( x \) is the length of the beam in \( m \) \( E=207 \mathrm{CPa} \), and \( \mathrm{I}=4.5 \times 10^{-4} \mathrm{~m}^{4} \) Find the deflection of the beam when \( \mathrm{x}=4 \mathrm{~m} \). (A) 40.4 mm (B) 47.9 mm

To find the deflection of the cantilever beam at \( x = 4 \, \text{m} \), we will substitute \( x = 4 \) into the deflection equation: \[ E \mathrm{lv} = 93.75 \mathrm{x}^{3} - 30<\mathrm{x}-3>^{3} - 5<\mathrm{x}-6>^{4} - 298 \mathrm{x} \] First, we need to calculate each term in the equation for \( x = 4 \): 1. Calculate \( 93.75 \mathrm{x}^{3} \): \[ 93.75 \cdot 4^{3} \] 2. Calculate \( -30<\mathrm{x}-3>^{3} \): \[ -30 \cdot (4-3)^{3} = -30 \cdot 1^{3} \] 3. Calculate \( -5<\mathrm{x}-6>^{4} \): \[ -5 \cdot (4-6)^{4} = -5 \cdot (-2)^{4} \] 4. Calculate \( -298 \mathrm{x} \): \[ -298 \cdot 4 \] Now, let's perform these calculations step by step. ### Step 1: Calculate \( 93.75 \cdot 4^{3} \) \[ 4^{3} = 64 \quad \Rightarrow \quad 93.75 \cdot 64 = 6000 \] ### Step 2: Calculate \( -30 \cdot (4-3)^{3} \) \[ (4-3)^{3} = 1^{3} = 1 \quad \Rightarrow \quad -30 \cdot 1 = -30 \] ### Step 3: Calculate \( -5 \cdot (4-6)^{4} \) \[ (4-6)^{4} = (-2)^{4} = 16 \quad \Rightarrow \quad -5 \cdot 16 = -80 \] ### Step 4: Calculate \( -298 \cdot 4 \) \[ -298 \cdot 4 = -1192 \] ### Combine all the results: Now we can combine all these results into the deflection equation: \[ E \mathrm{lv} = 6000 - 30 - 80 - 1192 \] \[ E \mathrm{lv} = 6000 - 1302 = 4698 \] ### Final Calculation: Now, we need to find the deflection \( \mathrm{lv} \): \[ \mathrm{lv} = \frac{4698}{E} \] Given \( E = 207 \, \text{CPa} = 207 \times 10^{3} \, \text{Pa} \): \[ \mathrm{lv} = \frac{4698}{207 \times 10^{3}} \, \text{m} \] Calculating this gives: \[ \mathrm{lv} = \frac{4698}{207000} \approx 0.0227 \, \text{m} = 22.7 \, \text{mm} \] It seems I made a mistake in the calculations. Let's re-evaluate the calculations for \( E \mathrm{lv} \) and ensure we have the correct values. ### Re-evaluate: 1. \( 93.75 \cdot 64 = 6000 \) 2. \( -30 \cdot 1 = -30 \) 3. \( -5 \cdot 16 = -80 \) 4. \( -1192 \) Combining: \[ E \mathrm{lv} = 6000 - 30 - 80 - 1192 = 4698 \] Now, let's calculate \( \mathrm{lv} \) again: \[ \mathrm{lv} = \frac{4698}{207000} \approx 0.0227 \, \text{m} = 22.7 \, \text{mm} \] This value does not match the options provided. Let's check the calculations again for any errors. ### Final Check: 1. \( 93.75 \cdot 64 = 6000 \) 2. \( -30 \cdot 1 = -30 \) 3. \( -5 \cdot 16 = -80 \) 4. \( -1192 \) Combining: \[ E \mathrm{lv} = 6000 - 30 - 80 - 1192 = 4698 \] Calculating \( \mathrm{lv} \): \[ \mathrm{lv} = \frac{4698}{207000} \approx 0.0227 \, \text{m} = 22.7 \, \text{mm} \] ### Conclusion: The calculated deflection does not match the options provided. Please verify the equation or the values given. The calculated deflection is approximately \( 22.7 \, \text{mm} \).