I go to the farm to buy more chickens. when I come home, I've multiplied the number of chickens i have by 7. I now have 42 chickens. How many chickens did I start with?
The question states that we have multiplied two numbers together to obtain a product number. we know one of the multiples, and we know the product. Let's write an equation to see what we know! ? chickens x 7 = 42 chickens Whatever we do to the left of the = we must do to the right in order to keep the equation true. Let's divide the left and right sides by 7, to reverse the multiplication! ? chickens x 7 = 42 chickens divide both sides by 7 ? chickens = 6 chickens Now we know I started with 6 chickens! Lets write our final answer. I had 6 chickens before I went to the farm!
Sabrina is moving along the x-axis on her way to school. Sabrina is often distracted, so he direction and velocity change quite often, in a pattern that, when graphing position, matches x(t)=(1/2)sin(t). Find the equation of Sabrina's velocity as well as her acceleration. Also find her instantaneous velocity at t=pi. For the sake of simplicity, all values of "C" can be assumed as 0.
To find the equation of a velocity curve, one simply must take the derivative of the position equation [x(t)=(1/2)sin(t)]. d/dt [(1/2)sin(t)] first, we can take out the constant because of the constant rule (1/2) d/dt sin(t) the derivative of sin is cos, so we are left with (1/2)cos(t) therefore the velocity equation is v(t)=(1/2)cos(t)+C C is assumed to be 0, so... v(t)=(1/2)cos(t) we repeat the same process to find the acceleration, however we are now taking the derivative of the velocity equation. d/dt [(1/2)cos(t)] (1/2) d/dt cos(t) the derivative of cos is -sin, so we are left with -(1/2)sin(t) therefore, the acceleration is a(t)=-(1/2)sin(t)+C C is assumed to be 0, so... a(t)=-(1/2)sin(t) Now we can solve the last part of the problem! When t=pi, What is Sabrina's velocity? v(pi)=(1/2)cos(pi) v(pi)=(1/2)(1) v(pi)=1/2 Let's write our final answer! v(t)=(1/2)cos(t) a(t)=-(1/2)sin(t) v(pi)=1/2
Sabrina needs to go to the store. She needs 1 carton of eggs, 3 loaves of bread, a box of cereal, and 2 gallons of milk. She has $43 to buy her groceries. She knows last time she went to the grocery store she bought one of each item above and spent $22. Sabrina also saw two specials that read "buy 2 cartons of eggs and one loaf of bread for $17," and "buy one loaf of bread and one gallon of milk for $8." Milk is always $3 at the store. Does Sabrina have enough money for her trip to the grocery store? If yes, how much will she have leftover? If no, how much more does she need?
This problem is a four variable system of equations. Let's begin by labeling each item with a letter variable. eggs "w" bread "x" cereal "y" milk "z" Now lets see what information is given. "She knows last time she went to the grocery store she bought one of each item above and spent $22" With this information we can write our first equation! w+x+y+z=22 What else? "Sabrina also saw two specials that read "buy 2 cartons of eggs and one loaf of bread for $17," and "buy one loaf of bread and one gallon of milk for $8." With this information we have two more equations! 2w+x=17 x+z=8 And lastly, "Milk is always $3 at the store." z=3 That is all of the given information! Let's try to solve for the unknown variables! w+x+y+z=22 2w+x=17 x+z=8 z=3 Since we know the value of z, lets replace the variable with the value! w+x+y+3=22 2w+x=17 x+3=8 Now we can solve for x in the last equation! x+3=8 subtract 3 from both sides x=5 Now let's replace all of the variable x with its value! w+5+y+3=22 2w+5=17 We can solve for w! 2w+5=17 subtract 5 from both sides 2w=12 divide both sides by 2 w=6 finally, lets replace w with 6! 6+5+y+3=22 let's combine like terms y+14=22 subtract 14 from both sides y=8 Yay! We found values for all of our variables! Let's take another look at the question to see what it is asking! "Does Sabrina have enough money for her trip to the grocery store? If yes, how much will she have leftover? If no, how much more does she need?" Let's remind ourselves of what Sabrina needs from her trip! "She needs 1 carton of eggs, 3 loaves of bread, a box of cereal, and 2 gallons of milk." What do we already know? w=6 x=5 y=8 z=3 We can write an equation to find the total cost of her trip! w+3x+y+2z=? Let's replace the variables with their corresponding values! 6+3(5)+8+2(3)=? Lets solve! 6+3(5)+8+2(3)=? multiply the numbers outside the parenthesis by the numbers inside 6+15+8+6=? add them up and we get.... 35! According to the problem Sabrina has $43, so it looks like she has enough money for the trip! Lets write our final answer! Sabrina has more than enough money to buy her groceries. The groceries will cost $35 total and Sabrina has $43, which means she will leave with $8 leftover.