The amount of money that a local charity earns by selling T-shirts at a mall depends on the price of each T-shirt. The monthly profit, 𝑦, in dollars is given by the quadratic equation 𝑦 = −35𝑥2 + 1250𝑥 − 6500 where 𝑥 represents the price of each T-shirt. Determine the vertex of the parabola
To find the vertex of the parabola, we need to find the value of 𝑥 that gives the maximum value of 𝑦. One way to do this is by using the formula 𝑥 = −𝑏/2𝑎, where 𝑏 is the coefficient of the linear term and 𝑎 is the coefficient of the quadratic term.
In this case, 𝑎 = −35 and 𝑏 = 1250. Therefore,
𝑥 = −1250/2(−35) = 1250/70 ≈ 17.86
This means that the maximum profit is obtained when the price of each T-shirt is approximately $17.86.
To find the maximum profit, we can substitute this value of 𝑥 into the equation 𝑦 = −35𝑥2 + 1250𝑥 − 6500:
𝑦 = −35(17.86)2 + 1250(17.86) − 6500 ≈ $11,056.96
Therefore, the vertex of the parabola is approximately (17.86, 11,056.96).
To determine the vertex of the parabola represented by the equation 𝑦 = −35𝑥^2 + 1250𝑥 − 6500, we need to use the formula for the x-coordinate of the vertex:
𝑥 = -𝑏 / (2𝑎),
where 𝑏 is the coefficient of the x term, and 𝑎 is the coefficient of the x^2 term.
In this case, 𝑎 = -35 and 𝑏 = 1250, so substituting these values into the formula, we have:
𝑥 = -1250 / (2*(-35))
= -1250 / (-70)
= 17.8571 (rounded to 4 decimal places)
Now we substitute this value into the equation to find the corresponding y-coordinate:
𝑦 = −35(17.8571)^2 + 1250(17.8571) − 6500
= -11250 + 22321.43 - 6500
= 4546.43 (rounded to 2 decimal places)
Therefore, the vertex of the parabola is approximately (17.8571, 4546.43).