Question 9 A budget of \( \$ 225 \) is available to buy steel sheeting to construct an open water tank in the shape of a rectangular prism of height \( h \mathrm{~cm} \), that is twice as long \( (2 x \mathrm{~cm}) \) as it is wide \( (x \mathrm{~cm}) \). Cut to size the sheeting costs 1.5 cents per square centimetre. (a) Show that the total cost of the steel, in dollars, is given by \( \frac{3 x^{2}+9 x h}{100} \).

To find the total cost of the steel sheeting required to construct the open water tank, we first need to determine the surface area of the rectangular prism (the tank) and then calculate the cost based on the price per square centimeter. 1. **Dimensions of the tank**: - Width = \( x \) cm - Length = \( 2x \) cm - Height = \( h \) cm 2. **Surface area of the rectangular prism**: The surface area \( A \) of a rectangular prism is given by the formula: \[ A = 2lw + 2lh + 2wh \] where \( l \) is the length, \( w \) is the width, and \( h \) is the height. Substituting the dimensions: - Length \( l = 2x \) - Width \( w = x \) - Height \( h = h \) The surface area becomes: \[ A = 2(2x)(x) + 2(2x)(h) + 2(x)(h) \] Now, calculate each term: - The area of the bottom and top (2lw): \[ 2(2x)(x) = 4x^2 \] - The area of the sides (2lh): \[ 2(2x)(h) = 4xh \] - The area of the front and back (2wh): \[ 2(x)(h) = 2xh \] Now, combine these areas: \[ A = 4x^2 + 4xh + 2xh \] \[ A = 4x^2 + 6xh \] 3. **Total surface area**: The total surface area \( A \) is: \[ A = 4x^2 + 6xh \] 4. **Cost of the steel sheeting**: The cost of the sheeting is given as 1.5 cents per square centimeter. To convert this to dollars, we note that 1.5 cents is \( \frac{1.5}{100} \) dollars, or \( 0.015 \) dollars. Therefore, the total cost \( C \) in dollars is: \[ C = \text{(cost per square cm)} \times \text{(surface area)} \] \[ C = 0.015 \times (4x^2 + 6xh) \] To express this in a more manageable form, we can factor out \( 0.015 \): \[ C = 0.015(4x^2 + 6xh) \] Now, to express this in terms of cents, we multiply by 100: \[ C = \frac{15(4x^2 + 6xh)}{100} \] \[ C = \frac{60x^2 + 90xh}{100} \] Finally, we can rewrite this as: \[ C = \frac{3x^2 + 9xh}{100} \] Thus, we have shown that the total cost of the steel, in dollars, is given by: \[ \frac{3x^2 + 9xh}{100} \]