Quadratic Equations in Real Life

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Answer explanation

The initial height of the ball is found by evaluating the height function at time t=0. Substituting t=0 into h(t) gives h(0) = -4.9(0)^2 + 20(0) + 1.5 = 1.5 meters. Thus, the initial height is 1.5 meters.

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A rectangular garden has a length that is 3 meters more than its width. If the area of the garden is 70 square meters, which quadratic equation represents this situation?

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Answer explanation

Let the width be x meters. Then, the length is x + 3 meters. The area is given by length × width = (x)(x + 3) = 70. This simplifies to the quadratic equation x(x + 3) = 70, making this the correct choice.

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Answer explanation

To find the maximum height, use the vertex formula for a parabola. The height function h(t) = -5t^2 + 30t has its maximum at t = -b/(2a) = 3 seconds. Plugging t = 3 into h(t) gives h(3) = 45 meters, the correct answer.

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The sum of a number and its square is 42. Which quadratic equation represents this situation?

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Answer explanation

The problem states that the sum of a number (x) and its square (x^2) equals 42. This can be expressed as x^2 + x = 42, making the first choice the correct quadratic equation.

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A rectangle has a perimeter of 30 meters. If the length is twice the width, which quadratic equation can be used to find the width?

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Answer explanation

The perimeter of a rectangle is given by 2(length + width). If the length is twice the width (length = 2x), then the equation becomes 2x + 2(2x) = 30, simplifying to 2x + 4x = 30, which is the correct choice.

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The product of two consecutive integers is 56. Which quadratic equation represents this situation?

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Answer explanation

The product of two consecutive integers can be expressed as x(x + 1). Setting this equal to 56 gives the equation x(x + 1) = 56, which is the correct representation of the situation.

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Answer explanation

The height function h(t) = -4.9t^2 + 100 indicates that the initial height (when t=0) is 100 meters. Therefore, the height of the cliff is 100 meters.

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The area of a square is 64 square meters. Which quadratic equation represents the side length of the square?

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Answer explanation

The area of a square is given by the formula A = x^2, where x is the side length. Given that the area is 64 square meters, we can set up the equation x^2 = 64, making this the correct choice.

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Answer explanation

The time taken by the faster train can be calculated using the formula time = distance/speed. The faster train's speed is (x + 10) km/h, so the equation is t = 100/(x + 10), which matches the correct choice.

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The height of a triangle is 4 meters less than its base. If the area of the triangle is 30 square meters, which quadratic equation represents this situation?

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Answer explanation

Let the base be x meters. The height is x - 4 meters. The area of a triangle is given by (1/2) * base * height. Thus, (1/2)x(x - 4) = 30 represents the situation, making the correct choice \(\frac{1}{2}x(x - 4) = 30\).