Solve Y=0.09(x-50)^3+15

$$Y = 0.09 ( x - 50 ) ^ { 3 } + 15$$

Solve for Y

$$Y=\frac{9\left(x-50\right)^{3}}{100}+15$$

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(x-50\right)^{3}.

$$Y=0.09\left(x^{3}-150x^{2}+7500x-125000\right)+15$$

Use the distributive property to multiply 0.09 by x^{3}-150x^{2}+7500x-125000.

$$Y=0.09x^{3}-13.5x^{2}+675x-11250+15$$

Add -11250 and 15 to get -11235.

$$Y=0.09x^{3}-13.5x^{2}+675x-11235$$

Solve for x (complex solution)

$$x=\sqrt[3]{\frac{100Y}{9}-\frac{500}{3}}+50$$

$$x=e^{\frac{i\times 4\pi }{3}}\sqrt[3]{\frac{100Y}{9}-\frac{500}{3}}+50$$

$$x=e^{\frac{i\times 2\pi }{3}}\sqrt[3]{\frac{100Y}{9}-\frac{500}{3}}+50$$

Solve for x

$$x=\sqrt[3]{\frac{100Y}{9}-\frac{500}{3}}+50$$