We are asked to sketch and label in inches three different designs for a measuring cup that holds 1 cup, which is about 14.4375 cubic inches. We could make this cup be a pyramid with some sophisticated base, like below.
However, then there will be a lot of computations to find the exact dimensions of such a cup. Let's make it simple and find measuring cups in the shape of a . We choose this shape because the volume of a rectangular prism is equal to the product of its dimensions, but keep in mind that there are many different solutions to this problem.
Option 1
First, let the measuring cup be in the shape of a cube and s be its side length. Then its volume is V=s^3. We know that the volume of the cup has to be about 14.4375 cubic inches. Let's find s!
V=s^3
14.4375=s^3
sqrt(14.4375)=s
s=sqrt(14.4375)
s≈ 2.43499
s≈ 2.43
Therefore, the side length of the cube should be about 2.43 inches. Now, let's sketch the cube!
Option 2
Now let the measuring cup be in the shape of a square prism with a side base length of 2 inches. Let h denote its height. Then its volume is V=2* 2* h=4h. We know that the volume of the cup has to be about 14.4375 cubic inches. Let's find h!
Therefore, the height of the prism should be about 3.61 inches. Now, let's sketch the prism!
Option 3
Now let the measuring cup be in the shape of a square prism with a side base length of 3 inches. Let h denote its height. Then its volume is V=3* 3* h=9h. We know that the volume of the cup has to be about 14.4375 cubic inches. Let's find h!
Therefore, the height of the prism should be about 1.6 inches. Now, let's sketch the prism!
<jsxgpre id="Solution41761_4" static=1>
var b = mlg.board([-2.5, 12.5, 18.5, -2.5], {desktopSize:'medium', style:'usa'});
/* %% Length of a Vector - Array */
function myLength(vec){
return Math.sqrt(vec[0]*vec[0]+vec[1]*vec[1]);
}
/* Crearing a right angle from three points - in Perspective Geometry is helpful*/
function myRightAngle(p_1,p_2,p_3,len){
var temp1X = p_2.X() + len*(p_1.X()-p_2.X())/myLength([p_2.X()-p_1.X(),p_2.Y()-p_1.Y()]);
var temp1Y = p_2.Y() + len*(p_1.Y()-p_2.Y())/myLength([p_2.X()-p_1.X(),p_2.Y()-p_1.Y()]);
var temp3X = p_2.X() + len*(p_3.X()-p_2.X())/myLength([p_2.X()-p_3.X(),p_2.Y()-p_3.Y()]);
var temp3Y = p_2.Y() + len*(p_3.Y()-p_2.Y())/myLength([p_2.X()-p_3.X(),p_2.Y()-p_3.Y()]);
var temp2X = temp1X+temp3X-p_2.X();
var temp2Y = temp1Y+temp3Y-p_2.Y();
var node_temp1 = b.node(temp1X,temp1Y);
var node_temp3 = b.node(temp3X,temp3Y);
var node_temp2 = b.node(temp2X,temp2Y);
b.segment(node_temp1,node_temp2,{strokeColor:'red',strokeWidth:1});
b.segment(node_temp3,node_temp2,{strokeColor:'red',strokeWidth:1});
}
/* %%%%%%%%%%% Find a geometric middle of an array of pts %%%%%%
%%%% Returns an array of two values %%%%% */
function myMiddle(p_arr){
var n_temp = p_arr.length;
var sumX = 0;
var sumY = 0;
for(var ii=0; ii<n_temp; ii++){
sumX = sumX + p_arr[ii].X();
sumY = sumY + p_arr[ii].Y();
}
return [sumX/n_temp, sumY/n_temp];
}
/* %%%%% create a segment between two points, type=0 gives a solid one, type=1 a dashed usual one (Zwierz) %%%% */
function mySegment(p_1,p_2,type){
if(type==0){ b.segment(p_1,p_2,{strokeWidth:1.5,dash:0});}
if(type==1){ b.segment(p_1,p_2,{strokeWidth:1,dash:2});}
}
/* %%%%% MEASURES %%%%%% */
function myMeasure(p_1,p_2,gap_,dist_,text_){
b.measure(p_1,p_2,{centerGap:gap_,legs:{strokeWidth:1,dash:2,length:dist_,strokeColor:'black'},label:{text:text_,anchor:0.5,mathMode:true,distance:0,rotate:false,position:'middle',precision:0,fontSize:0.9}});
}
var hi = 4.4-1.5;
var len = 1.2;
var p1 = b.node(0,0);
var p2 = b.node(4.8,0);
var p3 = b.node(4.8+2*len,len);
var p4 = b.node(2*len,len);
var q1 = b.node(p1.X(),p1.Y()+hi);
var q2 = b.node(p2.X(),p2.Y()+hi);
var q3 = b.node(p3.X(),p3.Y()+hi);
var q4 = b.node(p4.X(),p4.Y()+hi);
mySegment(p1,p2,0);
mySegment(p2,p3,0);
mySegment(p3,p4,1);
mySegment(p4,p1,1);
mySegment(q1,q2,0);
mySegment(q2,q3,0);
mySegment(q3,q4,0);
mySegment(q4,q1,0);
mySegment(p1,q1,0);
mySegment(p2,q2,0);
mySegment(p3,q3,0);
mySegment(p4,q4,1);
var front_face = b.polygon([p1,p2,q2,q1],{borders:{visible:false},fillColor:'#b3f0cb'});
var right_face = b.polygon([p2,p3,q3,q2],{borders:{visible:false},fillColor:'#b3f0c0'});
var top_face = b.polygon([q1,q2,q3,q4],{borders:{visible:false},fillColor:'#aabb00'});
b.gradient(front_face,{color:[mlg.tikzColor('darkorange!30'),"darkorange"],radial:true,cx:10,cy:10,r:100});
b.gradient(top_face,{color:[mlg.tikzColor('darkorange!30'),"darkorange"],radial:true,cx:10,cy:50,r:100});
b.gradient(right_face,{color:[mlg.tikzColor('darkorange!80'),"darkorange"],radial:true,cx:10,cy:50,r:100});
myMeasure(p1,p2,0.2,-1.5,'3');
myMeasure(p2,p3,0.4,-1.5,'3');
myMeasure(q1,p1,0.3,-2,'1.6');
myRightAngle(p4,p1,p2,0.7);
myRightAngle(p1,p2,p3,0.7);
myRightAngle(p2,p3,q3,0.7);
myRightAngle(p3,q3,q2,0.7);
b.txt(3.8,5,'(\\text{inches})');
b.trim();
</jsxgpre>
