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<p>A storage tank will have a circular base of radius <code class='latex inline'>r</code> and a height of <code class='latex inline'>r</code>. The tank can be either cylindrical or hemispherical (half sphere).</p><p>a. Write and simplify an expression for the ratio of the volume of the hemispherical tank to its surface area (including the base). For a sphere, V = 34—m3 and SA = 47rr2.</p><p>b. Write and simplify an expression for the ratio of the volume of the cylindrical tank to its surface area (including the bases).</p><p>c. Compare the ratios of volume to surface area for the two tanks.</p><p>d. Compare the volumes of the two tanks.</p><p>e. Describe how you used these ratios to compare the volumes of the two tanks. Which measurement of the tanks determines the volumes?</p>
Similar Question 2
<p>Fred drove his car a distance of <code class='latex inline'>2x km</code> in 3 h. Later, he drove a distance of <code class='latex inline'>x + 100</code> km in 2h. Use the equation speed = <code class='latex inline'>\frac{distance}{time}</code></p><p>Write a simplified expression for the difference between the fist speed and the second speed.</p>
Similar Question 3
<p>An object has mass <code class='latex inline'>\displaystyle m = \frac{p+1}{3p + 1}</code> and density <code class='latex inline'>\displaystyle \rho = \frac{p^2 - 1}{9p^2 + 6p + 1}</code>. Determine its volume <code class='latex inline'>v</code>, where <code class='latex inline'>\displaystyle \rho = \frac{m}{v}</code>. State the restrictions on any variables. </p>
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Learning Path
L1 Quick Intro to Factoring Trinomial with Leading a
L2 Introduction to Factoring ax^2+bx+c
L3 Factoring ax^2+bx+c, ex1
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<p>Newton&#39;s law of gravitation states that any two objects exert a gravitation al force on each other due to their masses, <code class='latex inline'>\displaystyle F_g = G\frac{m_1 m_2}{r^2}</code>, where <code class='latex inline'>F</code> is the gravitational force, <code class='latex inline'>G</code> is a constant (the universal gravitational constant), <code class='latex inline'>m_1</code> and <code class='latex inline'>m_2</code> are the masses of the objects, and <code class='latex inline'>r</code> is the separation distance between the centres of objects. The mass of Mercury is 2.2 times greater than the mass of Pluto. Pluto is 102.1 times as far from the Sun as Mercury. How many times greater is the gravitational force between the Sun and Mercury than the gravitational force between the Sun and Pluto?</p>
<p>A company purchases <code class='latex inline'>x</code> kilograms of steel for <code class='latex inline'>\$2249.52</code>. The company processes the steel and turns it into parts that can be used in other factories. After this process, the total mass of the steel has dropped by 25 kg (due to trimmings, scrap, and so on), but the value of the steel has increased to <code class='latex inline'>\$ 10, 838.52</code>. The company has made a profit of <code class='latex inline'>\$2/kg</code>. What was the original mass of the steel? What its the original cost per kilogram?</p> <p>The value, in thousands of dollars, of a certain new boat can be modelled by the equation <code class='latex inline'>V(t) = \frac{24}{t + 6}</code>, where <code class='latex inline'>t</code> is the time, in years.</p><p>a) Sketch a graph of this relation. State the domain and range for this situation.<br> b) What was the initial value of the boat?<br> c) What is the projected value of the boat after each time? </p> <ul> <li>i) 1 year<br></li> <li>ii) 3 years<br></li> <li>iii) 7 years<br></li> </ul> <p>Andrew and David are planning a canoe trip. They want to compare how the travel time for each portion of the trip will vary according to the speed at which they travel. For the first part of the trip, they will travel 24 km across a calm lake. For the second part of the trip, they will travel 18 km up a river whose current flows at 4 km/h. For the last part of the trip, they will travel 36 km down a river that flows at 3 km/h. they used the relationship <code class='latex inline'>t = \frac{d}{s}</code>, where <code class='latex inline'>t</code> is time in hours, <code class='latex inline'>d</code> is the distance in kilometres, and <code class='latex inline'>s</code> is the speed in kilometres per hour, to establish that at a speed of <code class='latex inline'>s</code> km/h, the time it will take to travel along the lake is <code class='latex inline'>t_{1} = \frac{24}{s}</code>, the time to travel up the river is <code class='latex inline'>t_2 = \frac{18}{s - 4}</code>, and the time taken to travel down the river is <code class='latex inline'>t_3 = \frac{36}{s + 3}</code>.</p><p>a) Identify the base function <code class='latex inline'>f(s)</code> for this situation.<br> b) Describe the transformations that must be applied to <code class='latex inline'>f(s)</code> to obtain each time function.<br> c) Graph each function on the same set of axes.<br> d) How long will each part of the trip take if they paddle their canoe at a constant speed of 6 km/h?</p> <p>Two cones have radii in the ratio 5:4 and heights in the ratio 2:3. Determine the ratio of their volumes, where <code class='latex inline'>\displaystyle V = \frac{1}{3}\pi r^2h</code>. </p><img src="/qimages/870" /> <p>Determine the area of the triangle in simplified form. State the restrictions.</p><img src="/qimages/871" /> <p>A rectangle is six times as long as it is wide. Determine the ratio of its area to its perimeter, in simplest form, if its width is <code class='latex inline'>w</code>. </p> <p>Determine the area of the triangle in simplified form. State the restrictions. </p><img src="/qimages/227" /> <p>Fred drove his car a distance of <code class='latex inline'>2x km</code> in 3 h. Later, he drove a distance of <code class='latex inline'>x + 100</code> km in 2h. Use the equation speed = <code class='latex inline'>\frac{distance}{time}</code></p><p>Determine the values of x for which the speed was greater of the second trip.</p> <p>For each function, find the inverse and the domain and range of the function and its inverse. Determine whether the inverse is a function.</p><p><code class='latex inline'>\displaystyle f(x)=3 x+4 </code></p> <p>Can two different rational expressions simplify to the same polynomials? Explain using examples.</p> <p>Explain why every polynomial is also a rational expression.</p> <p> Equation <code class='latex inline'>\displaystyle \frac{1}{x}+ \frac{1}{y} = \frac{1}{z}</code>, which can bet written as <code class='latex inline'>\displaystyle \frac{1}{x} = \frac{1}{z}- \frac{1}{y}</code>. Suppose you want to determine natural-number solutions of this equation; for example, <code class='latex inline'>\displaystyle \frac{1}{6} = \frac{1}{2} - \frac{1}{3}</code> and <code class='latex inline'>\displaystyle \frac{1}{20}= \frac{1}{4}- \frac{1}{5}</code>.</p><p>State two more solutions of the equation <code class='latex inline'>\displaystyle \frac{1}{x}= \frac{1}{z} - \frac{1}{y}</code>.</p> <p>Irene wants to make an open-topped box using a rectangular piece of cardboard with dimensions 120 cm by 100 cm. She plans to cut a square of side length <code class='latex inline'>x</code> from each corner.</p><img src="/qimages/5228" /><p>a) Write an expression for the volume of the box as a function of x.</p><p>b) Write an expression for the surface area of the open-topped box as a function of x.</p><p>c) State the domain for this situation.</p><p>d) Write a simplified expression for the ratio of the volume of the box to the surface area.</p><p>e) What are the restrictions on x?</p> <p>A storage tank will have a circular base of radius <code class='latex inline'>r</code> and a height of <code class='latex inline'>r</code>. The tank can be either cylindrical or hemispherical (half sphere).</p><p>a. Write and simplify an expression for the ratio of the volume of the hemispherical tank to its surface area (including the base). For a sphere, V = 34—m3 and SA = 47rr2.</p><p>b. Write and simplify an expression for the ratio of the volume of the cylindrical tank to its surface area (including the bases).</p><p>c. Compare the ratios of volume to surface area for the two tanks.</p><p>d. Compare the volumes of the two tanks.</p><p>e. Describe how you used these ratios to compare the volumes of the two tanks. Which measurement of the tanks determines the volumes?</p> <p>Jolon needs to fence a rectangular garden. The dimensions of the garden are 6 meters by 4 meters. How much fencrng does Jolon need to purchase?</p> <p>The width of the rectangle <code class='latex inline'>\frac{a+ 10}{3a + 24}</code>.</p><p>Write an expression for the length of the rectangle in the simplest form if the area of the rectangle is <code class='latex inline'> \displaystyle A = \frac{2a+20}{3a + 15} </code></p> <p>An object has mass <code class='latex inline'>\displaystyle m = \frac{p+1}{3p + 1}</code> and density <code class='latex inline'>\displaystyle \rho = \frac{p^2 - 1}{9p^2 + 6p + 1}</code>. Determine its volume <code class='latex inline'>v</code>, where <code class='latex inline'>\displaystyle \rho = \frac{m}{v}</code>. State the restrictions on any variables. </p> <p>Consider this rectangular swimming pool. <code class='latex inline'>\displaystyle A=432 \mathrm{ft}^{2} </code></p><p><code class='latex inline'>\displaystyle \ell=5 x-1 \mathrm{ft} </code></p><p><code class='latex inline'>\displaystyle A=432 \mathrm{ft}^{2} </code></p><p><code class='latex inline'>\displaystyle \ell=5 x-1 \mathrm{ft} </code> a) What does the expression <code class='latex inline'>\displaystyle \frac{432}{5 x-1} </code> represent in this situation?</p><p>b) State the domain and range for the expression in part a).</p><p>c) Does the expression in part a) represent a function? Justify your answer.</p><p>d) Determine the width of the pool when the length is <code class='latex inline'>\displaystyle 24 \mathrm{ft} </code>.</p> <p>Fred drove his car a distance of <code class='latex inline'>2x km</code> in 3 h. Later, he drove a distance of <code class='latex inline'>x + 100</code> km in 2h. Use the equation speed = <code class='latex inline'>\frac{distance}{time}</code></p><p>Write a simplified expression for the difference between the fist speed and the second speed.</p> <p>An isosceles triangle has two sides of length <code class='latex inline'>9x + 3</code>. The perimeter of the triangle is <code class='latex inline'>30x + 10</code>.</p><p><strong>(a)</strong> Determine the ratio of the base to the perimeter, in simplified form. State the restriction on <code class='latex inline'>x</code>.</p><p><strong>(b)</strong> Explain why the restriction on <code class='latex inline'>x</code> in part (a) is necessary in this situation.</p> <p>Mel is attending a very loud concert by The Restriced. To avoid permanent ear damage, he decides to move farther from the state. Sound intensity is given by the formula <code class='latex inline'>I = \frac{k}{d^2}</code>, where <code class='latex inline'>k</code> is a constant and <code class='latex inline'>d</code> is the distance in metres from the listener to the source of the sound. Determine an expression for the decrease in sound intensity if Mel moves <code class='latex inline'>x</code> metres farther from the stage.</p> <p>Kwok is a hotel manager. His responsibilities include renting rooms for conferences. The hotel charges$250 per day plus $15 per person for the grand ballroom.</p><p>Rearrange your formula to express <code class='latex inline'>n</code> in terms of <code class='latex inline'>C</code></p> <p>GROCERIES Mr. Bailey purchased some groceries that cost <code class='latex inline'>\displaystyle d </code> dollars. He paid with a <code class='latex inline'>\displaystyle \$ 50 </code> bill. Write an expression for the amount of change he will receive.</p>
<p>Consider a cylinder with height <code class='latex inline'>h = 3x - 4</code> and radius <code class='latex inline'>r = 2x + 3</code>.</p><img src="/qimages/5228" /><p>a) Determine the ratio of the volume of the cylinder to its surface area.</p><p>b) Determine any restrictions on <code class='latex inline'>x</code>.</p>
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