Anonymous
Anonymous asked in Science & MathematicsMathematics · 3 months ago

Can someone help me figure out these style of questions?

A rug is to fit in a room so that a border of even width is left on all four sides. If the room is 12 ft by 15 ft and the area of the rug is 108 sq ft, how wide will the border be?

A boat is 156 ft from the base of a cliff. If the distance from the top of the cliff to the boat is 39 ft more than twice the height of the cliff, what is the height of the cliff? Round to the nearest tenth if a foot, if necessary.

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    (12 - 2x) * (15 - 2x) = 108

    x is the width of the border

    2 * (6 - x) * (15 - 2x) = 108

    (6 - x) * (15 - 2x) = 54

    90 - 12x - 15x + 2x^2 = 54

    2x^2 - 27x + 36 = 0

    x = (27 +/- sqrt(729 - 8 * 36)) / 4

    x = (27 +/- sqrt(9 * (81 - 8 * 4))) / 4

    x = (27 +/- 3 * sqrt(81 - 32)) / 4

    x = (27 +/- 3 * sqrt(49)) / 4

    x = (27 +/- 3 * 7) / 4

    x = (27 +/- 21) / 4

    x = 48/4 , 6/4

    x = 12 , 1.5

    x = 12 doesn't make sense in the original problem

    x = 1.5

    The rug measures 9x12 feet

    The next one is a right triangle, so you'll need the Pythagorean theorem

    156^2 + h^2 = (2h + 39)^2

    156^2 + h^2 = 4h^2 + 156h + 39^2

    0 = 3h^2 + 156h + 39^2 - 156^2

    39^2 - 156^2 =>

    (39 - 156) * (39 + 156) =>

    39 * (1 - 4) * 39 * (1 + 4) =>

    39^2 * (-3) * 5 =>

    -15 * 39^2

    0 = 3h^2 + 156h + 39^2 - 156^2

    0 = 3h^2 + 156h - 15 * 39^2

    0 = h^2 + 52h - 5 * 39^2

    h = (-52 +/- sqrt(52^2 + 4 * 5 * 39^2)) / 2

    h = (-52 +/- sqrt(4^2 * 13^2 + 4 * 5 * 3^2 * 13^2)) / 2

    h = (-52 +/- 13 * 2 * sqrt(4 + 5 * 3^2)) / 2

    h = (-52 +/- 26 * sqrt(4 + 5 * 9)) / 2

    h = (-52 +/- 26 * sqrt(49)) / 2

    h = (-52 +/- 26 * 7) / 2

    h > 0, presumably

    h = (26 * 7 - 52) / 2

    h = 26 * (7 - 2) / 2

    h = 13 * 5

    h = 65

  • 3 months ago

    The style is to convert words to algebra and then solve.

    1. If the border has width x, then the dimensions of the rug are (12 - 2x) by (15 - 2x), and the area is the product of those numbers.

    (12 - 2x)(15 - 2x) = 108

    180 - 54x + 4x^2 = 108

    4x^2 - 54 + 72 = 0

    2x^2 - 27 + 36 = 0

    To factor that, you need numbers that multiply to 2*36=72 and add up to -27. A quick check shows that (-3)(-24) works, so:

    2x^2 - 3x - 24x + 36 = 0 ... rewrite -27 as -3 - 24 in the middle term

    (2x - 3)x - 12(2x - 3) = 0 ... factor out GCF in pairs

    (2x - 3)(x - 12) = 0

    So x = 12 or x = 3/2 will solve the equation, but x must be less than half the short side of the room, so 12 is too big. The border must be 3/2 = 1.5 ft. wide.

    If you didn't see that right away, it would help to sketch a picture. This one would be a 12x15 rectangle (the room) with a slightly smaller rectangle (the rug) inside. Each side of the smaller rectangle is x feet from the closest wall of the room.

    2. Sketch some more. You have have a right triangle and I'd label that with a boat at B, the top of the cliff at A and the base of the cliff at C. Read the problem and you find that side a is 156, side b is the unknown height of the cliff, and are told that side c is 2b + 39.

    Now use the Pythagorean theorem:

    156² + b² = (2b + 39)²

    Expand the square on the right and simplify. You'll get another quadratic equation to solve.

    In both problems, the best tool in your belt (besides a brain :^) is a pencil and paper to sketch the situation.

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