Word problemsChapter 7: Practice/review problemsThe collection of problems listed below contains questions taken from previous MA123 exams.Max-min problems[1]. A field has the shape of a rectangle with two s emicircles attached at opposite sides. Find the radius ofthe semicircles if the field is to have maximum area, the perimeter of the field equals 100, and the widthof the field (twice the radius of the semicircles) is at most 18. (Caution: Be sure your answer satisfiesall conditions.) The radius equals(a) 6 (b) 7 (c) 8 (d) 9 (e) 10[2]. Find the area of the largest rectangle with one corner at the origin and the opposite corner in the firstquadrant and on the line y = 10 − 2x. Assume the sides of the rectangle are parallel with the axes.(a) 73/2 (b) 67/2 (c) 55/2 (d) 49/2 (e) 25/2[3]. If you sell an item at price p, your revenue will equal the price p times the number sold, n. Suppose priceis linearly related to the number sold by the equ ationn = 100 − 10(p − 10)How s hould you set the price to maximize revenue? The price should equal(a) 10 (b) 15 (c) 20 (d) 25 (e) 30[4]. A rectangle in the fi rst quadrant has one corner at (0, 0) and the opposite corner on the curve y = 2 −x2.What is the largest possible area of this rectangle?(a)23r73(b)83r23(c)83r43(d)43r23(e)23r43[5]. Find th e length of the shortest line segment that connects the point (4, 0) in the (x, y) plane to the liney = 2x.(a)85√5 (b)107√7 (c)1617√17 (d)1215√15 (e)1819√19[6]. Find the area of the triangle of minimum area with base equal to the unit interval 0 ≤ x ≤ 1 on the xaxis and with opposite vertex lying on the curve y = 8x +4x2with x > 0.(a) 1 (b) 2 (c) 3 (d) 4 (e) 6103[7]. Find th e area of the largest rectangle with one corner at the origin, the opposite corner in the firstquadrant on the graph of the line f(x) = 6 − 3x, and sides parallel to the axes.(a) 1 (b) 2 (c) 3 (d) 4 (e) 5[8]. What is the maximum area of the rectangle with sides parallel to the coordin ate axes, one corner at theorigin, and the opposite corner in the first quadrant on the ellipse given by the equation 2x2+ y2= r2?(a) r2(b)r2√2(c)r22(d)r22√2(e)r24[9].A rectangular field as shown below is constructed using 2400feet of fencing. (There are six parallel fences in the verticaldirection.) What is the maximum possible area in square feetof the rectangular field?(a) 100, 000 (b) 110, 000 (c) 120, 000 (d) 130, 000 (e) None of the above[10].Find the point (x0, y0) in th e first quadrant that lieson the hyperbola y2− x2= 5 and is closest to th epoint A(4, 0). Then (x0, y0) is(a) (1,√6)(b) (2, 3)(c) (2.5,√11.25)(d) (3,√14)(e) (4,√21)xy0•A[11]. Suppose you want to find the shortest distance between the point (1, 0) on the x-axis and a point on theellipse x2+ 4y2= 16. Which problem do you need to solve?(a) Minimize D =vuut(x − 1)2+ r16 − x24!2where − 4 ≤ x ≤ 4.(b) Minimize D =vuut(x)2+ r16 − x24− 1!2where −4 ≤ x ≤ 4.(c) Minimize D =r(x)2+p16 − 4x2− 12where −2 ≤ x ≤ 2.(d) Minimize D =r(x − 1)2+p16 − 4x22where −2 ≤ x ≤ 2.(e) None of the above.104[12]. Suppose y =32x2. What is the minimum sum of x and y if x and y are both positive?(a) 6 (b) 9 (c) 3 (d) 2 (e) 4[13]. Suppose that the sum of x and y is 12, x and y both positive. What is the value of x that gives thelargest possible value of x2y?(a) 6(b)√6(c) 8(d)√8(e) 4[14]. Suppose th e product of x and y is 64 and both x and y are positive. What is the minimum possible sumof x and y?(a) 9 (b) 12 (c) 15 (d) 16 (e) 20[15]. Find the area of the rectangle of maximum area with one vertex (corner) at (0, 0) and opposite corner onthe ellipse x2+ 4y2= 4.(a) 3/4(b)√5/4 (c)√7/4(d) 1(e)√11/4[16]. Let T be the triangle enclosed by the x-axis, the y-axis, and the line y = 4 − 2x. Find the area of thelargest rectangle with sides parallel to the coordinate axes that can be inscribed in T .(a) 2 square units (b) 8 square units (c) 4 square units(d) 6 square units (e) 3 square units[17]. Let (a, b) be the point on the line y = 4 −2x that is closest to the origin (0, 0). What is the distance from(a, b) to (0, 0)? (Hint: Draw a picture.)(a) 2√5/5 (b) 3√5/5 (c) 4√5/5 (d) 5√5/5 (e) 6√5/5Related rate problems[18]. At 12:00 noon a ship sailing due East at 20 miles per hour passes directly North of a lighthouse located onthe coast exactly one mile South of the ship. How fast is the distance between the ship and the lighthouseincreasing at 1:00 pm?(a)100√101(b)200√201(c)300√301(d)400√401(e)500√501[19]. Water is evaporating at a rate of .5 cubic feet per day from a cylindrical tank. The circular base of thetank (parallel to the ground) has a radius of 4 feet. How fast is the depth of the water decreasing whenthe tank is half full (measured in feet per day)?(a)164π(b)132π(c)116π(d)18π(e)14π105[20]. A triangle has a base of length 5 on the x axis. The altitude of the triangle is increasing at a rate of 3units per s econd. How fast is the area of the triangle increasing when the area of the triangle equals 14square units?(a)452(b)402(c)352(d)252(e)152[21]. A train travels along a straight track at a constant speed of 50 miles per hour. A straight road intersectsthe track at right angles and a truck is parked on the road one mile from the track. How fast is the tr aintraveling away fr om the truck when the train is 3 miles past the intersection?(a) 10√10 (b) 15√10 (c) 20√10 (d) 5√10 (e) 20√5[22]. Water is evaporating from a pool at a constant rate. The area of the pool is 5000 square feet. Assumethe sides of the pool drop straight down (perpendicular) from th e edge. Th e water in the pool drops .5feet in one day. How fast is the water evaporating in cubic feet per day?(a) 2000 (b) 2500 (c) 3000 (d) 3500 (e) 4000[23]. A point moves along the line y = 4 + 3x so that the y coordinate of the point increases at a constant rateof 2 units per second. How fast is the x coordinate of the point increasing?(a) 2/3 (b) 1 (c) 3/2 (d) 2 (e) 3[24]. Two trains leave a station at the s ame time. One travels north on a track at 30 miles per hour. Thesecond travels east on a track at 40 miles per hour. How fast are the trains travelling away from eachother in miles per hour when the northbound train is 60 miles from the station?(a) 60 miles per hour (b) 40 miles per hour (c) 50 miles per hour(d) 130 miles per hour(e) 50√5 m iles per hour[25]. A sandbox with square base is being fi lled with sand at th e