Math III Section 4.6 #29 and #31


The photo above gives evidence about Descartes Rule; a technique for determining the number of positive or negative real roots of a polynomial. Three negative polynomials become 1 negative because two negatives become a positive BUT with another added negative, this will make the following polynomial a negative. The highlighted polynomial is an incorrect polynomial because two negatives become one positive as shown above.

The following information above shows the step in how to get the dimensions needed with the following given:  "A gardener is designing a new garden in the shape of a trapezoid. She wants the shorter base to be twice the height and the longer base to be 4 feet longer than the shorter base. If she has enough topsoil to create a 60ft^2 garden, what dimensions should she use for the garden?" First you have to know the area for a trapezoid before you begin, which is (1/2)(h)(b1+b2). After, you then begin to plug in the information you. Base 1 is twice the height (2h), Base 2 is 4 feet longer than the short base (4+2h) and the height is equal (h=h). After, you then plug in your information. 60ft^2= (1/2)(h)(2h+(2h+4)). The next to the last step before finishing your equation you subtract 60 from the left to get it to equal "0". The final and last step to completing your equation is to combine like terms.. 0=(1/2)(h)(4h+4)-60.

The photo above represents the equation Y=(1/2)(h)(4h+4)-60. After plugging the following equation into your graphic cal you then use the table to get the dimension of a part of the base. "5" represents "h".

When finalizing your entire problem you just plug 5 "h" into all the places you identify the letter "h". Base 1 is "2h". (2(5))= 10. Base 2 is "4+2h". (4+2(5))=14 . After identifying what Base 1 and 2 are, you then plug your following information into the original equation; 60=(1/2)(h)(b1+b2).....60=(1/2)(5)(10+14). After plugging your information into your original equation you then just multiply and add . (1/2)(5)= 2.5 (10+14)= 24. After multiplying and adding the grouped numbers you then just multiply the following numerals together (2.5)(24)= 60. Since, the multiplied factor is equal to the original 60ft then this is the correct dimensions.  

The dimensions the Gardener should use for the Trapezoid Garden are the following:

Base 1: 10ft

Base 2: 14ft

Height: 5

60=(1/2)(5)= 2.5 ... (10+14)=24

✓ 60=60

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