Question Video: Solving a Quadratic Equation in a Worded Problem Using the Quadratic Formula Mathematics

A stone is thrown upward from the top of a cliff and lands in the sea some time later. The height, ℎ, above sea level at any time, 𝑡 seconds, is given by the formula ℎ = 3𝑡 − 5𝑡² + 40. After how many seconds will the stone reach the sea? Give your answer to the nearest tenth of a second.

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Video Transcript

A stone is thrown upward from the top of a cliff and lands in the sea some time later. The height ℎ above sea level at any time 𝑡 seconds is given by the formula ℎ is equal to three 𝑡 minus five 𝑡 squared plus 40. After how many seconds will the stone reach the sea? Give your answer to the nearest tenth of a second.

In this question, we are given a quadratic equation for ℎ in terms of 𝑡. Rewriting the right-hand side in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐, we have ℎ is equal to negative five 𝑡 squared plus three 𝑡 plus 40. We are trying to calculate the time at which the stone will reach the sea, and we are told that ℎ is the height above sea level. This means that when the stone reaches the sea, ℎ will be equal to zero. And we need to solve the quadratic equation negative five 𝑡 squared plus three 𝑡 plus 40 equals zero. We know that any quadratic equation 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero can be solved using the quadratic formula. This states that 𝑥 is equal to negative 𝑏 plus or minus the square root of 𝑏 squared minus four 𝑎𝑐 all divided by two 𝑎.

In this question, our values of 𝑎, 𝑏, and 𝑐 are negative five, three, and 40, and the variable instead of being 𝑥 is the time 𝑡. Substituting in our values, we have 𝑡 is equal to negative three plus or minus the square root of three squared minus four multiplied by negative five multiplied by 40 all divided by two multiplied by negative five. This simplifies to negative three plus or minus the square root of 809 all divided by negative 10. Separating our two solutions, we have 𝑡 is equal to negative three plus the square root of 809 divided by negative 10 or 𝑡 is equal to negative three minus the square root of 809 divided by negative 10.

Typing these into the calculator, we have 𝑡 is equal to negative 2.544 and so on and 𝑡 is equal to 3.144 and so on. As the time in seconds must be a positive value, we can rule out the first answer. We are also asked to give the time to the nearest tenth of a second. We can therefore conclude that it takes 3.1 seconds for the stone to reach the sea.

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