Skateboard Revenue:
A skateboard shop sells about 50 skateboards per week for the price advertised. For each $1 decrease in price, about 1 more skateboard per week is sold. The shop's revenue can be modeled by:
\[ y = (70 - x)(50 + x) \]
Use vertex form to find how the shop can maximize weekly revenue.
Answer (2)
The question pertains to finding the maximum weekly revenue for a skateboard shop where the revenue function is given by y=(70-x)(50+x). To find the maximum value, we first put the quadratic equation into vertex form, which is y=a(x-h)^2+k, where (h, k) is the vertex of the parabola. The x-coordinate of the vertex, which is 'h' in the vertex form, gives us the price adjustment that will maximize revenue. The y-coordinate 'k' will be the maximum revenue itself.
To convert y=(70-x)(50+x) to vertex form, we complete the square: y = 70×50 + 70x - x×50 - x×x y = 3500 + 20x - x^2 y = -x^2 + 20x + 3500
Now, we complete the square: y = -(x^2 - 20x) + 3500 y = -(x^2 - 20x + 100) + 3500 + 100 y = -(x - 10)^2 + 3600
This is now in vertex form, where the vertex is (10, 3600). This means the shop can maximize weekly revenue by decreasing the price by $10, which will yield a maximum revenue of $3600.
To maximize weekly revenue, the skateboard shop should decrease its price by $10, leading to a maximum revenue of $3600. This conclusion was reached by converting the revenue function into vertex form and identifying the vertex. The vertex point indicates the optimal price adjustment for maximum revenue.
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