You are able to purchase one of the two different types of hay bales: cylinder shaped or rectangular shaped. The cylinder hay bale is 4 feet high and has a diameter of 6 feet. The rectangular hay bale has a width of 2 feet, height of 1 ½ feet, and length of 3 feet. Note: You must purchase whole bales. 1. Approximately how many rectangular shaped bales would you need to purchase to have the same amount of hay as one cylinder shaped bale? 2. You need 16,000 ft3 of hay to last through the year. Determine how many of each hay bale you would need to purchase if you purchase either all rectangular bales or all cylinder shaped bales. If one cylinder shaped bale costs $20 and one rectangular bale costs $2.75, which will be the better value? 3. Your barn measures 36 feet long, 36 feet wide, and 12 feet high. What is the greatest number of cylinder shaped bales that will fit in the barn? rectangular bales? Which type of hay bale will allow you to fit a greater volume of hay in the barn? *PLease Show Work*

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Answer:     the interval on which the curve is concave up is  (-∞, -1/2)======    we find the interval where this curve is concave up by finding the interval in which the 2nd derivative is positive.differentiate the curve equation with respect to x    let the function f denote be the anti-derivative of the inside expression, i.e. a function such that .then use the fundamental theorem of calculus:     differentiate again to get second derivative. we can use chain rule with 1/x as the outside function and 1 + x + x² as the inside function    find critical numbers of the second derivative. these numbers are where the 2nd deriv. is undefined or zero.where the 2nd derivative is zero: set (d²y/dx²) to equal 0 and solve for x    where the 2nd derivative is undefined: if the denominator is equal to zero, the derivative will be undefeined. attempting to solve the equation    1 + x + x² = 0results in a discriminant of      meaning that the equal 1 + x + x² = 0 has no real solutions and therefore will never equal zero.======our only critical number is x = -1/2.this forms two intervals to test the sign for: (-∞, -1/2) and (-1/2, ∞)test values into the 2nd derivative for (-∞, -1/2)if we try x = -1, we get      so the since the 2nd derivative of the curve is positive on the interval  (-∞, -1/2), the curve is concave up on (-∞, -1/2)test values into the 2nd derivative for (-1/2, ∞)if we try x = 0, we get      so the since the 2nd derivative of the curve is positive on the interval  (-∞, -1/2), the  curve is concave down on (-∞, -1/2).the interval on which the curve is concave up is  (-∞, -1/2)