Question

select the correct answer. engineers are designing a cylindrical sensor for scientific equipment used in space exploration. the surface area of the sensor must be no more than 100.48 square centimeters and the sensor is 6 cm long. what is the maximum radius of the sensor? 1 cm 2 cm 8 cm 4 cm

Explanation:

Step1: Recall cylinder surface - area formula

The surface - area formula of a cylinder is $S = 2\pi r^2+2\pi rh$, where $r$ is the radius and $h$ is the height. Given $S\leq100.48$ square centimeters and $h = 6$ cm. So $S=2\pi r^2 + 2\pi r\times6=2\pi r^2+12\pi r$.

Step2: Substitute $\pi\approx3.14$

We get $2\times3.14r^2+12\times3.14r\leq100.48$. Simplify it to $6.28r^2 + 37.68r-100.48\leq0$. Divide through by $6.28$ to obtain $r^2 + 6r - 16\leq0$.

Step3: Solve the quadratic inequality

Factor the quadratic equation $r^2 + 6r - 16=(r + 8)(r - 2)$. The roots of the equation $r^2+6r - 16 = 0$ are $r=-8$ and $r = 2$ from $r+8 = 0$ and $r - 2=0$. Since the radius $r>0$, and considering the inequality $r^2 + 6r - 16\leq0$, the valid range for $r$ is $0

Answer:

2 cm