multiple choice 2 points\nin an effort to catch a criminal; a superhero is going to leap over a building and…

multiple choice 2 points\nin an effort to catch a criminal; a superhero is going to leap over a building and take a short cut down the ally. the function f(x)=−16x² + 128x gives the superheros height in feet as a function of time. the building is 425 feet high. will the superhero make it over the building?\nyes, the superhero will make it over the building because the superhero can leap a max of 478 feet.\nno, the superhero will not make it over the building because the superhero can only leap a max of 256 feet.\nmultiple choice 2 points\na fireworks rocket is launched from a hill above a lake. the rocket will fall into the lake after exploding at its maximum height. the rockets height above the lake is given by f(x)=−16x² + 64x + 80. how long will it take the rocket to hit the lake?\n5 seconds\n8 seconds\n10 seconds\n12 seconds

multiple choice 2 points\nin an effort to catch a criminal; a superhero is going to leap over a building and take a short cut down the ally. the function f(x)=−16x² + 128x gives the superheros height in feet as a function of time. the building is 425 feet high. will the superhero make it over the building?\nyes, the superhero will make it over the building because the superhero can leap a max of 478 feet.\nno, the superhero will not make it over the building because the superhero can only leap a max of 256 feet.\nmultiple choice 2 points\na fireworks rocket is launched from a hill above a lake. the rocket will fall into the lake after exploding at its maximum height. the rockets height above the lake is given by f(x)=−16x² + 64x + 80. how long will it take the rocket to hit the lake?\n5 seconds\n8 seconds\n10 seconds\n12 seconds

Answer

Explanation:

Step1: Find the maximum of $f(x)=-16x^{2}+128x$

The function $y = ax^{2}+bx + c$ has its vertex (maximum for $a<0$) at $x=-\frac{b}{2a}$. Here $a=-16$, $b = 128$. So $x=-\frac{128}{2\times(-16)}=\frac{-128}{-32}=4$.

Step2: Calculate the maximum height

Substitute $x = 4$ into $f(x)=-16x^{2}+128x$. Then $f(4)=-16\times4^{2}+128\times4=-16\times16 + 512=-256+512 = 256$ feet. Since $256<425$, the superhero will not make it over the building.

Step3: For the rocket - find when it hits the lake

Set $f(x)=-16x^{2}+64x + 80=0$. Divide through by -16 to get $x^{2}-4x - 5=0$.

Step4: Factor the quadratic equation

Factor $x^{2}-4x - 5=(x - 5)(x+1)=0$.

Step5: Solve for $x$

Using the zero - product property, $x - 5=0$ or $x+1=0$. So $x = 5$ or $x=-1$. Since time $x\geq0$, the rocket hits the lake at $x = 5$ seconds.

Answer:

No, the superhero will not make it over the building because the superhero can only leap a max of 256 feet. A. 5 seconds