This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Fresh day Ntsako, let's solve. Question 2.1: Concrete Shaft First, we need to find the height (h) of the cone. The problem states the slant height is 15 cm. The diagram shows a radius of 8 cm associated with this slant height. The problem also states the inside radius is 8 cm. Therefore, we will use the inside radius and the given slant height to find the common height of the shaft. Using the Pythagorean theorem (h^2 + r^2 = l^2): h^2 + (8 cm)^2 = (15 cm)^2 h^2 + 64 cm^2 = 225 cm^2 h^2 = 225 cm^2 - 64 cm^2 h^2 = 161 cm^2 h = sqrt(161) cm The formula for the volume of a cone is V = (1)/(3) r^2 h. 2.1.1 Calculate the volume of the shaft using the inside radius Step 1: Identify the inside radius and height. Inside radius r_in = 8 cm. Height h = sqrt(161) cm. Step 2: Substitute the values into the volume formula. V_in = (1)/(3) (8 cm)^2 (sqrt(161) cm) V_in = (1)/(3) (64) sqrt(161) cm^3 V_in = 64sqrt(161)3 cm^3 Step 3: Calculate the numerical value. V_in ≈ (64 × 12.688577 × 3.14159265)/(3) cm^3 V_