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MATH 111: HW 4 SOLUTIONS AND COMMENTS
4.1 #3. The Pythagorean theorem states that in any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. In the right triangle of this problem, we therefore have
hypotenuse^2 = 1^2 + x^2
and so hypotenuse =
1 + x^2.
4.1 #4. We know one side of the rectangle has length 4 inches; to find the area of the rectangle, we want find the length of the other side of the rectangle.
The diagonal divides the rectangle into two right triangles. Both of those triangles have hypotenuse 5 inches and one side of length 4 inches. By the Pythagorean theorem,
52 = 4^2 + (other side)^2 ,
so (other side)^2 = 25 โ 16 = 9
and therefore (other side) = 3 inches.
Therefore, the rectangle has sides of length 3 inches and 4 inches, and the area of the rectangle is 3 inches times 4 inches: that is, the area is 12 square inches.
4.1 #6. The angles sum to 180 degrees (a straight angle).
4.1 #8. No, there cannot be a right triangle with sides of length 1, 2, 3. If there were such a triangle, the hypotenuse (always the longest side) would have length 3. Then the Pythagorean theorem would require that the equation
12 + 2^2 = 3^2
be correct.
But, we know that 1^2 + 2^2 does not equal 3^2 (because 1 + 4 does not equal 9!!) So there cannot be a right triangle with those side lengths.
4.1 #15. The first step is to note that 1 mile is 5280 feet. (If you didnโt know this already, this is easy to look up on the internet.) In the problem, the track expands to 5280+2 = 5282 feet.
The two halves of the track are each right triangles, with base 5280/2 = 2640 feet and hypotenuse 5282/2 = 2641 feet. The height of the midpoint of the track is therefore
height^2 = 2641^2 โ 26402 = 5281
and so height =
5281 โ 72 .67 feet.
4.3 #3. The ratio of two sides of a Golden rectangle is
ฯ =
The ratios of the sides of the rectangles in the problem are: 5/ 3 โ 1 .666; 11/ 8. 5 โ 1 .294; 14 / 11 โ 1 .273; 17/ 11 โ 1 .545. The closest of these to 1.618 is the first, so the index card most closely resembles a Golden rectangle.
4.3 #4. The two equations have the same solutions because they are essentially the same equation: the second equation is gotten from the first equation by taking the reciprocal of both sides. Therefore, any number which solves the first equation will solve the second, and vice-versa.
4.3 #5. Recall the quadratic formula: if we want to solve the equation ax^2 + bx + c = 0 for x, then there are two solutions:
x =
โb ยฑ
b^2 โ 4 ac 2 a
Now letโs look at the particular equations in this problem.
(1) Cross-multiplying yields 2x(x โ 1) = 1, which expands to 2x^2 โ 2 x = 1. Bring the 1 to the other side of the equation: 2x^2 โ 2 x โ 1 = 0. Now apply the quadratic formula (in this case a = 2, b = โ2, c = โ1) to see that the two solutions are
x =
Caution! Notice that
โ 3; so it is NOT the case that 12 /4 =
(2) Cross-multiplying yields x(x โ 4) = 2 ยท 3, which expands to x^2 โ 4 x = 6. Bring the 6 to the other side of the equation: x^2 โ 4 x โ 6 = 0. Now apply the quadratic formula (in this case a = 1, b = โ4, c = โ6) to see that the two solutions are
x =
(3) Cross-multiplying yields 3x(x + 1) = 2, which expands to 3x^2 + 3x = 2. Bring the 2 to the other side of the equation: 3x^2 + 3x โ 2 = 0. Now apply the quadratic formula (in this case a = 3, b = 3, c = โ2) to see that the two solutions are
x =
4.3 #9. Yes, the new smaller rectangle is a Golden rectangle. The sides of the new rectangle are half the length of the sides of original rectangle; but this means that the ratio of the sides of the new rectangle is exactly the same as the ratio of the sides of the original rectangle.
