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d=3.2(t+1)(2t-3) An air pump increases the oxygen levels in an aquarium and reduces the build-up of waste materials. The equation shown gives the depth, d, in inches of an air bubble beneath the surface of the water t seconds after it emerges from the air pump. After how many seconds does the air bubble reach the surface?

d=3.2(t+1)(2t−3) d=3.2(t+1)(2 t-3) \newlineAn air pump increases the oxygen levels in an aquarium and reduces the build-up of waste materials. The equation shown gives the depth, d d , in inches of an air bubble beneath the surface of the water t t seconds after it emerges from the air pump. After how many seconds does the air bubble reach the surface?

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Q. d=3.2(t+1)(2t−3) d=3.2(t+1)(2 t-3) \newlineAn air pump increases the oxygen levels in an aquarium and reduces the build-up of waste materials. The equation shown gives the depth, d d , in inches of an air bubble beneath the surface of the water t t seconds after it emerges from the air pump. After how many seconds does the air bubble reach the surface?
  1. Understand the Problem: Understand the problem.\newlineWe are given an equation d=3.2(t+1)(2t−3)d=3.2(t+1)(2t-3) that represents the depth of an air bubble in an aquarium as a function of time tt. We need to find out when the air bubble reaches the surface. The surface is represented by a depth of 00 inches.
  2. Set Depth to 00: Set the depth dd to 00 to find when the bubble reaches the surface.\newlineWe need to solve the equation 0=3.2(t+1)(2t−3)0=3.2(t+1)(2t-3) for tt.
  3. Factor Out Constant: Factor out the constant 3.23.2. Since 3.23.2 is a constant multiplier, we can divide both sides of the equation by 3.23.2 to simplify our equation. This gives us 0=(t+1)(2t−3)0=(t+1)(2t-3).
  4. Apply Zero Product Property: Apply the zero product property.\newlineThe zero product property states that if a product of two factors is zero, then at least one of the factors must be zero. Therefore, we set each factor equal to zero: t+1=0t+1=0 and 2t−3=02t-3=0.
  5. Solve First Factor: Solve the first factor for tt. Solving t+1=0t+1=0 for tt gives us t=−1t=-1.
  6. Solve Second Factor: Solve the second factor for tt. Solving 2t−3=02t-3=0 for tt gives us 2t=32t=3, which simplifies to t=32t=\frac{3}{2} or t=1.5t=1.5.
  7. Evaluate Solutions: Evaluate the solutions.\newlineThe solution t=−1t=-1 does not make sense in this context because time cannot be negative. Therefore, we discard this solution. The solution t=1.5t=1.5 seconds is the time when the air bubble reaches the surface.

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