(Solved): A 0.3kg model rocket is loaded with 0.85kg of rocket fuel. After launch, the rocket rises at ...

A $$0.3?\mathrm{kg}$$ model rocket is loaded with $$0.85\mathrm{kg}$$ of rocket fuel. After launch, the rocket rises at a constant rate of $$4\mathrm{m}/\mathrm{s}$$, but the rocket fuel is dissipated at a rate of $$0.15\mathrm{kg}/\mathrm{s}$$. Find the work done in propelling the rocket $$20\mathrm{m}$$ above the ground. (Use $$9.8\mathrm{m}/{\mathrm{s}}^2$$ for g.) Find the work (in J) needed to lift the model rocket itself. (Ignore the work needed to lift the fuel.) J Show how to approximate the required work (in J) to lift the rocket fuel by a Riemann sum. (Let $$x$$ be the distance in meters above the ground. Enter $$x_{\mathrm{i}}^{?}$$ as $$x_{\mathrm{i}}$$.) $$\underset{n??}{\lim }\underset{i=1}{\overset{n}{?}}\left(8.33?0.735x_i\right)?x$$ Express the work (in J) required to lift the rocket fuel as an integral of $$x$$ (in $$\mathrm{m}$$ ). $${?}_0^1((x)dx$$ Evaluate the integral (in J). $$\times 3$$ Find the total work (in J) by adding the work of lifting the model rocket to the work of lifting the rocket fuel. $$x$$ J [-/12.5 Points] SCALCET9 6.4.006. The table shows values of a force function $$f(x)$$, where $$x$$ is measured in meters and $$f(x)$$ in newtons. Use the midpoint rule with $$n=4$$ to estimate the work $$W$$ (in $$\mathrm{J}$$ ) done by the force in moving an object from $$x=3$$ to $$x=35$$. $$w=\mathrm{J}$$ [0/12.5 Points] SCALCET9 6.4.007.MI. A force of $$14\mathrm{lb}$$ is required to hold a spring stretched 2 in. beyond its natural length. How much work $$W$$ is done in stretching it from its natural length to 7 in. beyond its natural length? $$W=\text{ ft- }\mathrm{lb}$$ [-/12.5 Points] SCALCET9 6.4.012.EP. According to Hooke's Law, the force required to hold the spring stretched $$x\mathrm{m}$$ beyond its natural length is given by $$f(x)=kx$$, where $$k$$ is the spring constant. Suppose that $$12\mathrm{J}$$ of work is needed to stretch a spring from $$14\mathrm{cm}$$ to $$18\mathrm{cm}$$ and another $$20\mathrm{J}$$ is needed to stretch it from $$18\mathrm{cm}$$ to $$22\mathrm{cm}$$. Find the exact value of $$k$$, in $$\mathrm{N}/\mathrm{m}$$. $$k=$$ $$\mathrm{N}/\mathrm{m}$$ What is the natural length of the spring, in $$\mathrm{cm}$$ ? $$\mathrm{cm}$$