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I load my paper that includes 8 questions. Please read and follow the direction and if question instruction mention to graph direction, please do algebraically and graph both. Please, type your answer.
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Week 7 HW
1
Section 12.4.4:
3. Here we look at the factorization of cubic polynomials.
a) Using ideas from calculus, explain why the graph of an odd-degree polynomial with
real coefficients must cross the horizontal axis.
b) Without appealing to the Fundamental Theorem of Algebra, explain why a cubic
polynomial with real coefficients must factor into a product of a linear polynomial
and a quadratic polynomial (both with real coefficients).
Section 13.1.3:
13. Mrs. D. draws a large right triangle for her students, and shows that the two legs are 3
and 4 feet long, and the hypotenuse is 5 feet long. She computes that the area is 6 square
feet. As an aside, she remarks that using inches, the side lengths would be 36, 48, and 60
inches. Jimmie wonders whether the area of the triangle would be 72 square inches since
12 × 6 = 72 (but he’s afraid to ask). Is he correct? Explain. (Does your explanation feature a
calculation or a drawing?)
Week 7 HW
2
Section 13.2.3:
5. Investigating Scale Changes in Two Dimensions.
a) Construct two rectangles that are similar.
b) Determine the ratio of the perimeter of the larger rectangle to the perimeter of the
smaller rectangle.
c) Determine the ratio of the area of the larger rectangle to the area of the smaller
rectangle.
d) Compare the ratios of the linear factors (e.g., the sides lengths) of the two rectangles,
the perimeter of the two rectangles, and the area of the two rectangles.
e) Repeat parts (a)-(d) for pair of similar triangles and a pair of circles.
f) Based on your work above, form a conjecture about the ratio of the permeters and
the ratio of the areas of two-dimensional similar figures.
g) Prove your conjecture in the case of two similar rectangles.
Week 7 HW
3
24. How independent are area and perimeter? For certain geometric figures (e.g., circles,
squares, and equilateral triangles), area and perimeter convey the same information. This
happens because a single number (radius of a circle, side length of a square or equilateral
triangle) determines the figure up to congruence.
a) Find an equation that relates the area and the perimeter (circumference) of a circle.
(Express each in terms of the radius r , then use algebra to eliminate r.)
b) Find an equation that relates the area and perimeter of a square.
c) Find an equation that relates the area and perimeter of an equilateral triangle.
Section 13.3.4:
9. There are several parallelograms that have (0, 0), (4, 1), and (2, 3) as vertices. Find all of
them (by finding the fourth vertex). Sketch them. Are the parallelograms all congruent? Do
they all have the same area? Discuss.
Week 7 HW
4
14. If C = A + B, what does Heron’s formula give for the area of the triangle? Is this the
correct answer?
Section 13.4.5:
6. Measuring the perimeter of the Koch snowflake. Suppose the length of a side of the initial
triangle used to construct the Koch snowflake is 1.
a) Determine the length of the curve at each of the first four stages (stages 0 to 3).
b) Construct a formula for the length of the curve at the nth stage.
c) Find the limit of this length as n → ∞ . (It is reasonable to call this the perimeter of
the Koch snowflake.)
Week 7 HW
5
7. Measuring the area of the Koch snowflake. Again, suppose the length of a side of the
initial triangle used to construct the Koch snowflake is 1.
a) Determine the area of the region enclosed by the curve at stages 0 to 3.
b) Construct a formula for the area of the region enclosed by the curve at the nth
stage.
c) Find the area of the region enclosed by the Koch snowflake.
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