Astring or rope will break apart if it is placed under too much tensile stress. thicker ropes can withstand more tension without breaking because the thicker the rope, the greater the cross-sectional area and the smaller the stress. one type of steel has density 7890 kg/m3 and will break if the tensile stress exceeds 7.0×108n/m2. you want to make a guitar string from a mass of 4.0 g of this type of steel. in use, the guitar string must be able to withstand a tension of 900 n without breaking. your job is to determine (a) the maximum length and minimum radius the string can have; (b) the highest possible fundamental frequency of standing waves on this string, if the entire length of the string is free to vibrate.

The monthly demand equation for an electric utility company is estimated to be p equals 60 minus left parenthesis 10 superscript negative 5 baseline right parenthesis x, where p is measured in dollars and x is measured in thousands of killowatt-hours. the utility has fixed costs of $3 comma 000 comma 000 per month and variable costs of $32 per 1000 kilowatt-hours of electricity generated, so the cost function is upper c left parenthesis x right parenthesis equals 3 times 10 superscript 6 baseline plus 32 x. (a) find the value of x and the corresponding price for 1000 kilowatt-hours that maximize the utility's profit. (b) suppose that the rising fuel costs increase the utility's variable costs from $32 to $38, so its new cost function is upper c 1 left parenthesis x right parenthesis equals 3 times 10 superscript 6 baseline plus 38 x. should the utility pass all this increase of $6 per thousand kilowatt-hours on to the consumers?