Answers Investigation 1

Transcription

1 Applications. Students ma use various sketches. Here are some eamples including the rectangle with the maimum area. In general, squares will have the maimum area for a given perimeter. Long and thin rectangles will have a smaller area. This is a principle that students have encountered in earlier units of CMP, but it ma be a surprising result. A = A = A = A = Students ma use a table to verif that a square has the maimum area of m with side lengths of m. Encourage students to take bigger increments for the base. Then students can use this to estimate where the maimum point occurs. The table can also be used to sketch a graph. Students ma use the trace button on their graphing calculator to find the maimum point. X X= Y Students ma put their sketches on graph paper to verif the areas. The rectangle with the greatest area for this fied perimeter has sides all of which are. X= Y=. As in Eercise, sketches of possible rectangles ma var. Students ma choose to consider a table or graph to analze the situation and find that the maimum area is,. when the sides are both.. When making the table, encourage students to take bigger increments for the base. The table can be used to estimate where the maimum point occurs and sketch a graph. A =, A = X... X=. X=. A = Y..... Y=. A =,.. Note: At this point, students might not use graphing calculators, so their graphs will be sketched on paper. After Problem., the students can revisit Eercises and to find the equations. The equations are helpful in finding the tables or graphs. In Eercise, the equation is A = /( - /), and in Eercise, the equation is A = /( - /). Frogs, Fleas, and Painted Cubes Investigation

2 . a. Possible answer: The graph first increases and then decreases. It has reflectional smmetr at =.. It crosses the -ais at (, ) and (, ). b.. cm ; the rectangle has a base and width length of. cm. c. No such minimum area rectangle eists. If we find a rectangle with a given fied perimeter and a small area, we can alwas find another rectangle with the same perimeter and an even smaller area. This process of finding smaller and smaller areas can continue indefinitel. Note: If we propose that one of the dimensions is zero, then the area becomes zero and it is no longer a rectangle. d. cm e. This can be found b using a point on the graph. For eample, the point (, ) represents a rectangle with an area of cm and dimensions cm b cm. Because the dimensions are and, the fied perimeter is cm. I could also look at the greatest area,, on the graph. The greatest area of this graph is at =.. Because the greatest area alwas represents a square, the fied perimeter is *. =.. Length (cm). Width (cm). Area (cm ). a. In the table, the maimum area,. cm, is for the side length of. cm. This point, (.,.), is in the middle of the range of values for length in the table and is the highest point of the graph. As the length of a side increases from. to, the area decreases from. to. This increase and then decrease can be seen in the Area column in the table as it is shown in the shape of graph. b. Looking for the maimum area in the table, I need to find the middle of the range of values for length in the table or where the area starts to decrease. The -value (area) of this point is the maimum area. For an point, I divide the area (-value) b the length (-value) to find the width. Frogs, Fleas, and Painted Cubes Investigation

3 Area. a. Possible answer: The graph first increases and then decreases. It has reflectional smmetr at =. It crosses the -ais at (, ) and (, ). The maimum -value is. b. m ; the length and width are both. c. m ; m ; these two rectangles are related because the have the same dimensions and area, but the length and width are switched. d. The dimensions are m b m e. m; if the length is m, the area is m, so the width is m. So since P = (/ + w), the perimeter is ( + ) = m. Students might take advantage of the observation in part (c).. a. As the length of a side increases b, the area increases first, and then it decreases after the length of a side is more than. b. m c. The shape of the graph is a parabola that opens down. Rectangles With a Perimeter of Length of Side d. Possible approimate dimensions:. m b. m e. The dimensions should be m b m.. a. The graph is a parabola that crosses the -ais at (, ) and (, ). It has a greatest point at the point (, ). Note: Students ma choose to use their graphing calculators to sketch this graph and use the trace button or make a table to obtain various characteristics from the graph. b. The highest point on the graph is at =. The corresponding -value is. For this reason, the maimum area is m and the dimensions are m b m. c. A = /( - /) A = ( - ) A = () = m d. Using the dimensions of the rectangle with the maimum area, the fied perimeter P = ( + ) =. I can also find this from the equation, where is the sum of the dimensions of the rectangles. Multipling the sum b, the fied perimeter m.. a. w = - / b. A = /( - /) c. The graph goes up and then down with the shape of an upside-down U, or a parabola. It has reflectional smmetr at =.. = ( ) d. A = /( - /) A = ( - ) A = () = m e. First look for the length on the -ais, and then go up until ou hit the curve. From there, go across to the -ais. At this point, the -value or area is m. Frogs, Fleas, and Painted Cubes Investigation

4 = ( ) f. I go down the Length column until I get to, and then go across to find the area of m. g. To find the maimum area, students can either use a table, graph, or a trace on their calculator. The maimum area is. m and the dimensions are. m b. m.. a. - / b. A = /( - /). c. The graph is a parabola that opens down. It has reflectional smmetr at =.. = ( ) d. A = /( - /) A = ( - ) A = () = m e. First look for the length on the -ais, and then go up until ou hit the curve. Go across to the -ais values. At this point, the -value or area is m. = ( ) Area (m ) f. Go down in the Length column until ou get to m, and then go across to find the area of m. g. To find the maimum area, students can either use a table, graph, or trace button on their calculator. The maimum area is. m with dimensions. m b. m.. a. Students ma use smmetr to complete their graphs with the additional points and then fill in the curve. Rectangles With a Fied Perimeter Length (m) b. Rectangles With Perimeter of m Length (m) Area (m ) c. The maimum area is m with dimensions of m-b- m.. C Frogs, Fleas, and Painted Cubes Investigation

5 . a. Rectangles With Lengths Greater Than b. Area Length (m) Area (m ) Rectangles With Perimeters of Length c. The dimensions are m-b- m.. F. a. Profits of a Photographer Profit Sales Price $ $ $ $ $ $ $ $ $ $ $ $, $, $, $, Profit $ $ $, $, $, $, $, $, S, $ $ Photographer Profits $ $ $ $ $ $ $ $ Sales Price b. The price with the most profit is +. The highest point in the graph is (, ). In the table, the profit column shows maimum amount, +,, at the sales price of +. c. The shape of the graph is the same. As the -value increases, the -value increases at first and then decreases in both the table and the graph. The equation is also the same form, but with different numbers. In Problem. the equation was A = /( - /), while in this Eercise, the equation has instead. Frogs, Fleas, and Painted Cubes Investigation

6 Connections. The rectangle with dimensions of length and has the least perimeter of centimeters. Student can make a table to find the least perimeter. Rectangles With an Area of. D Length Width Perimeter. a. ( + ) =, m or () + () =, +, =, m. b. The Distributive Propert states that if two numbers are multiplied together and one is a sum, then the other factor can be distributed over the sum. If a, b, and c are numbers, then the Distributive Propert states that: a(b + c) = ab + ac. The Distributive Propert also states that if each number in a sum has a common factor, then the common factor can be factored out from each number and the sum can be written as a product. The area of a rectangle that has been subdivided into two rectangles can be calculated b multipling the length and width of the original rectangle or b calculating the area of the smaller rectangles and adding them. Note: Eercises are a review of the Distributive Propert from Accentuate the Negative. The Distributive Propert will be etended in the net Investigation to quadratic epressions.. ( + ) = () + () = + =. ( + ) = () + () = + =. ( - ) = () - () = - =. ( + ) = () + () = + =. + = ( + ). + = ( + ). + = ( + ) or ( + ) or ( + ) or ( + ). =. =. As increases b one unit, increases b units; the graph of the equation is a straight line with a slope of and a -intercept of. In the table as increases b one unit, increases b units.. As increases b one unit, decreases b units; the graph of the equation is a straight line with a slope of - and -intercept of. In the table as increases b one unit, the -values are decreasing b units.. As is increase b one unit, increases b an times the prior -value. The graph of the equation slopes upward, and increases faster as increases. In the table, as increases b, the -values are times the prior difference. This is an eponential function.. As increases b one unit, decreases. The decrease in -values is fast for values of near to zero, and slower as is further from zero. Because is valid for all values of, there is no corresponding -value for =. On the graph, the curve gets closer and closer to the -ais from the left, but will never cross the -ais. The same pattern appears in the table for positive and negative values of.. a. If w represents the width of the field and the length is / = - w, then the perimeter of the fields is P = [( - w) + w] = ards, which is the perimeter given. b. No, this is a linear relation with negative slope. As the width increases, the length decreases. c. Yes; the lengths of opposite sides of a parallelogram are equal. The perimeter is ards, so half the perimeter is ards. For this reason, / + w = or / = - w. Frogs, Fleas, and Painted Cubes Investigation

7 d. No; a quadrilateral could have sides of different lengths. For eample, a trapezoid could have at least one pair of opposite sides that aren t equal in length.. a b. Rectangles With an Area of, ft Length (ft) Width (ft).. Perimeter (ft).. c. According to the table above, the column of perimeter decreases first, and then increases after the length of one side is greater than. The rectangles with greater difference of length and width have large perimeters. The rectangles with lesser difference of length and width have small perimeters. d. / =, w Etensions. a. To obtain the maimum area of m, the dimensions of the rectangle should be m b m. Students can make a table to find the maimum area. It should reflect that the perimeter of m onl includes three sides of the rectangle. That is, / + w = (or / + w = ). Rectangles With a Three-sided Perimeter of (m) Length Width Area Frogs, Fleas, and Painted Cubes Investigation

8 b. The shape and area of both rectangles with the maimum areas are different. In part (a), the dimensions are m b m with m of area while in Problem., the dimensions are m b m with m of area. c. Both graphs have the same parabolic shape with a maimum point, which indicates the greatest area. However, the maimum points are different. Area O Length Frogs, Fleas, and Painted Cubes Investigation